Finding the number of subgroups of $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2} $.












4












$begingroup$



What is the number of subgroups of $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2} $?




I'm not really sure how to figure it out. I tried seeing subgroups of each $Bbb Z_{p^n}$ but I'm not sure I'm going to get them all.






abstract-algebra group-theory finite-groups p-groups







share|cite|improve this question











$endgroup$












  • $begingroup$
    The Sylow theorems tell me it exists and that there is only in $mathbb{Z}_{p^2}$, i dont know how to use them for the other one. A group of order p^2 isnt a Sylow-p subggroup in there.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 12:23






  • 1




    $begingroup$
    Sylow's theorems will not help you with this question. They do not tell you anything about groups of prime power order. This question is not particularly easy, but you should be able to make a start. For example, can you determine how many subgroups there are of order $p$?
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:28






  • 1




    $begingroup$
    OK, so if there are $n$ elements of order $p$, then there are $n/(p-1)$ subgroups of order $p$. Also note that all elements of order $p$ are contained in the unique subgroup that is isomorphic to $Z_p oplus Z_p$.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:47






  • 1




    $begingroup$
    Please don't change the question like this.
    $endgroup$
    – Shaun
    Dec 30 '18 at 13:28






  • 1




    $begingroup$
    I changed the question back to the original to avoid confusion.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 14:15
















4












$begingroup$



What is the number of subgroups of $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2} $?




I'm not really sure how to figure it out. I tried seeing subgroups of each $Bbb Z_{p^n}$ but I'm not sure I'm going to get them all.






abstract-algebra group-theory finite-groups p-groups







share|cite|improve this question











$endgroup$












  • $begingroup$
    The Sylow theorems tell me it exists and that there is only in $mathbb{Z}_{p^2}$, i dont know how to use them for the other one. A group of order p^2 isnt a Sylow-p subggroup in there.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 12:23






  • 1




    $begingroup$
    Sylow's theorems will not help you with this question. They do not tell you anything about groups of prime power order. This question is not particularly easy, but you should be able to make a start. For example, can you determine how many subgroups there are of order $p$?
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:28






  • 1




    $begingroup$
    OK, so if there are $n$ elements of order $p$, then there are $n/(p-1)$ subgroups of order $p$. Also note that all elements of order $p$ are contained in the unique subgroup that is isomorphic to $Z_p oplus Z_p$.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:47






  • 1




    $begingroup$
    Please don't change the question like this.
    $endgroup$
    – Shaun
    Dec 30 '18 at 13:28






  • 1




    $begingroup$
    I changed the question back to the original to avoid confusion.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 14:15














4












4








4


4



$begingroup$



What is the number of subgroups of $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2} $?




I'm not really sure how to figure it out. I tried seeing subgroups of each $Bbb Z_{p^n}$ but I'm not sure I'm going to get them all.






abstract-algebra group-theory finite-groups p-groups







share|cite|improve this question











$endgroup$





What is the number of subgroups of $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2} $?




I'm not really sure how to figure it out. I tried seeing subgroups of each $Bbb Z_{p^n}$ but I'm not sure I'm going to get them all.






abstract-algebra group-theory finite-groups p-groups




abstract-algebra group-theory finite-groups p-groups






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 30 '18 at 14:29









Henning Makholm

241k17307546




241k17307546










asked Dec 30 '18 at 12:19









Pedro SantosPedro Santos

1458




1458












  • $begingroup$
    The Sylow theorems tell me it exists and that there is only in $mathbb{Z}_{p^2}$, i dont know how to use them for the other one. A group of order p^2 isnt a Sylow-p subggroup in there.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 12:23






  • 1




    $begingroup$
    Sylow's theorems will not help you with this question. They do not tell you anything about groups of prime power order. This question is not particularly easy, but you should be able to make a start. For example, can you determine how many subgroups there are of order $p$?
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:28






  • 1




    $begingroup$
    OK, so if there are $n$ elements of order $p$, then there are $n/(p-1)$ subgroups of order $p$. Also note that all elements of order $p$ are contained in the unique subgroup that is isomorphic to $Z_p oplus Z_p$.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:47






  • 1




    $begingroup$
    Please don't change the question like this.
    $endgroup$
    – Shaun
    Dec 30 '18 at 13:28






  • 1




    $begingroup$
    I changed the question back to the original to avoid confusion.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 14:15


















  • $begingroup$
    The Sylow theorems tell me it exists and that there is only in $mathbb{Z}_{p^2}$, i dont know how to use them for the other one. A group of order p^2 isnt a Sylow-p subggroup in there.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 12:23






  • 1




    $begingroup$
    Sylow's theorems will not help you with this question. They do not tell you anything about groups of prime power order. This question is not particularly easy, but you should be able to make a start. For example, can you determine how many subgroups there are of order $p$?
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:28






  • 1




    $begingroup$
    OK, so if there are $n$ elements of order $p$, then there are $n/(p-1)$ subgroups of order $p$. Also note that all elements of order $p$ are contained in the unique subgroup that is isomorphic to $Z_p oplus Z_p$.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 12:47






  • 1




    $begingroup$
    Please don't change the question like this.
    $endgroup$
    – Shaun
    Dec 30 '18 at 13:28






  • 1




    $begingroup$
    I changed the question back to the original to avoid confusion.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 14:15
















$begingroup$
The Sylow theorems tell me it exists and that there is only in $mathbb{Z}_{p^2}$, i dont know how to use them for the other one. A group of order p^2 isnt a Sylow-p subggroup in there.
$endgroup$
– Pedro Santos
Dec 30 '18 at 12:23




$begingroup$
The Sylow theorems tell me it exists and that there is only in $mathbb{Z}_{p^2}$, i dont know how to use them for the other one. A group of order p^2 isnt a Sylow-p subggroup in there.
$endgroup$
– Pedro Santos
Dec 30 '18 at 12:23




1




1




$begingroup$
Sylow's theorems will not help you with this question. They do not tell you anything about groups of prime power order. This question is not particularly easy, but you should be able to make a start. For example, can you determine how many subgroups there are of order $p$?
$endgroup$
– Derek Holt
Dec 30 '18 at 12:28




$begingroup$
Sylow's theorems will not help you with this question. They do not tell you anything about groups of prime power order. This question is not particularly easy, but you should be able to make a start. For example, can you determine how many subgroups there are of order $p$?
$endgroup$
– Derek Holt
Dec 30 '18 at 12:28




1




1




$begingroup$
OK, so if there are $n$ elements of order $p$, then there are $n/(p-1)$ subgroups of order $p$. Also note that all elements of order $p$ are contained in the unique subgroup that is isomorphic to $Z_p oplus Z_p$.
$endgroup$
– Derek Holt
Dec 30 '18 at 12:47




$begingroup$
OK, so if there are $n$ elements of order $p$, then there are $n/(p-1)$ subgroups of order $p$. Also note that all elements of order $p$ are contained in the unique subgroup that is isomorphic to $Z_p oplus Z_p$.
$endgroup$
– Derek Holt
Dec 30 '18 at 12:47




1




1




$begingroup$
Please don't change the question like this.
$endgroup$
– Shaun
Dec 30 '18 at 13:28




$begingroup$
Please don't change the question like this.
$endgroup$
– Shaun
Dec 30 '18 at 13:28




1




1




$begingroup$
I changed the question back to the original to avoid confusion.
$endgroup$
– Derek Holt
Dec 30 '18 at 14:15




$begingroup$
I changed the question back to the original to avoid confusion.
$endgroup$
– Derek Holt
Dec 30 '18 at 14:15










2 Answers
2






active

oldest

votes


















1












$begingroup$

The given group is $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$. We will use the formula $$ |(m,n)|=lcm(|m|,|n|),$$ for any $ m in mathbb{Z}_{p^3}$ and $n in mathbb{Z}_{p^2}$, where $|.|$ denote the order .



Since $m,n$ are the elements of $mathbb{Z}_{p^3}$ and $mathbb{Z}_{p^2}$ respectively, we have $ |m| $ divides $ p^3$ and $|n|$ divides $p^2$.



(i) Show that there is only one subgroup of order of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_p oplus mathbb{Z}_p$.



In this step we calculate the number of element $(m,n)$ of order $p$. You easily check that there are $p^2-1$ elements having order $p^2$. All these $p^2-1$ elements along with the identity element $(0,0)$ form a unique subgroup of order $p^2$ that is isomorphic to $mathbb{Z}_p oplus mathbb{Z}_p$.



(ii) Show that there are $p^2+p$ subgroup of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_{p^2}$.



In this step we will count the number of cyclic subgroups of order $p^2$. In other words we have to find all elements $(m,n)$ of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ having order $p^2$. To do this we need to find



$(a)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|m|=p^2 in mathbb{Z}_{p^3}$,



$(b)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|n|=p^2 in mathbb{Z}_{p^2}$



From part $(a)$ and part $(b)$ Try to show that there are total $p^4-p^2$ elements of order $p^2$ and each cyclic group order $p^2$ has $p^2-p$ elements of order $p^2$.



Hence the number of cyclic subgroup of order $p^2$ is equal to $frac{p^4-p^2}{p^2-p} =p^2+p$.



Thus there is a total of $p^2+p+1$ subgroups of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ of order $p^2$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Alright Thanks, But my problem is that i dont know how to prove those statements.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:04










  • $begingroup$
    You only seem to be counting the subgroups of order $p^2$. There are others.
    $endgroup$
    – Jeremy Rickard
    Dec 30 '18 at 13:11












  • $begingroup$
    I only wanted those of order p^2, still dont know how to do it though
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:13










  • $begingroup$
    @PedroSantos, I have calculated the result for subgroup of order $p$2$ only
    $endgroup$
    – M. A. SARKAR
    Dec 30 '18 at 13:15










  • $begingroup$
    Yes i know thanks, but im not sure how you did it .
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:16



















1












$begingroup$

To answer the complete question it is helpful to know that, in a finite abelian group, the number of subgroups of order $n$ is equal to the number of index $n$ for any $n$. This is basically because finite abelian groups $G$ are isomorphic to their dual groups ${rm Hom}(G,{mathbb C}^times)$, and subgroups of the group correspond to quotients of the dual.



So the final answer is $2(1 + (p+1) + (p^2+p+1)) = 2p^2+4p+6$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 17:09






  • 1




    $begingroup$
    That follows from the fact that finite abelian groups are direct sums of cyclic groups.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 18:07











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056775%2ffinding-the-number-of-subgroups-of-mathbbz-p3-oplus-mathbbz-p2%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

The given group is $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$. We will use the formula $$ |(m,n)|=lcm(|m|,|n|),$$ for any $ m in mathbb{Z}_{p^3}$ and $n in mathbb{Z}_{p^2}$, where $|.|$ denote the order .



Since $m,n$ are the elements of $mathbb{Z}_{p^3}$ and $mathbb{Z}_{p^2}$ respectively, we have $ |m| $ divides $ p^3$ and $|n|$ divides $p^2$.



(i) Show that there is only one subgroup of order of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_p oplus mathbb{Z}_p$.



In this step we calculate the number of element $(m,n)$ of order $p$. You easily check that there are $p^2-1$ elements having order $p^2$. All these $p^2-1$ elements along with the identity element $(0,0)$ form a unique subgroup of order $p^2$ that is isomorphic to $mathbb{Z}_p oplus mathbb{Z}_p$.



(ii) Show that there are $p^2+p$ subgroup of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_{p^2}$.



In this step we will count the number of cyclic subgroups of order $p^2$. In other words we have to find all elements $(m,n)$ of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ having order $p^2$. To do this we need to find



$(a)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|m|=p^2 in mathbb{Z}_{p^3}$,



$(b)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|n|=p^2 in mathbb{Z}_{p^2}$



From part $(a)$ and part $(b)$ Try to show that there are total $p^4-p^2$ elements of order $p^2$ and each cyclic group order $p^2$ has $p^2-p$ elements of order $p^2$.



Hence the number of cyclic subgroup of order $p^2$ is equal to $frac{p^4-p^2}{p^2-p} =p^2+p$.



Thus there is a total of $p^2+p+1$ subgroups of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ of order $p^2$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Alright Thanks, But my problem is that i dont know how to prove those statements.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:04










  • $begingroup$
    You only seem to be counting the subgroups of order $p^2$. There are others.
    $endgroup$
    – Jeremy Rickard
    Dec 30 '18 at 13:11












  • $begingroup$
    I only wanted those of order p^2, still dont know how to do it though
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:13










  • $begingroup$
    @PedroSantos, I have calculated the result for subgroup of order $p$2$ only
    $endgroup$
    – M. A. SARKAR
    Dec 30 '18 at 13:15










  • $begingroup$
    Yes i know thanks, but im not sure how you did it .
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:16
















1












$begingroup$

The given group is $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$. We will use the formula $$ |(m,n)|=lcm(|m|,|n|),$$ for any $ m in mathbb{Z}_{p^3}$ and $n in mathbb{Z}_{p^2}$, where $|.|$ denote the order .



Since $m,n$ are the elements of $mathbb{Z}_{p^3}$ and $mathbb{Z}_{p^2}$ respectively, we have $ |m| $ divides $ p^3$ and $|n|$ divides $p^2$.



(i) Show that there is only one subgroup of order of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_p oplus mathbb{Z}_p$.



In this step we calculate the number of element $(m,n)$ of order $p$. You easily check that there are $p^2-1$ elements having order $p^2$. All these $p^2-1$ elements along with the identity element $(0,0)$ form a unique subgroup of order $p^2$ that is isomorphic to $mathbb{Z}_p oplus mathbb{Z}_p$.



(ii) Show that there are $p^2+p$ subgroup of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_{p^2}$.



In this step we will count the number of cyclic subgroups of order $p^2$. In other words we have to find all elements $(m,n)$ of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ having order $p^2$. To do this we need to find



$(a)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|m|=p^2 in mathbb{Z}_{p^3}$,



$(b)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|n|=p^2 in mathbb{Z}_{p^2}$



From part $(a)$ and part $(b)$ Try to show that there are total $p^4-p^2$ elements of order $p^2$ and each cyclic group order $p^2$ has $p^2-p$ elements of order $p^2$.



Hence the number of cyclic subgroup of order $p^2$ is equal to $frac{p^4-p^2}{p^2-p} =p^2+p$.



Thus there is a total of $p^2+p+1$ subgroups of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ of order $p^2$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Alright Thanks, But my problem is that i dont know how to prove those statements.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:04










  • $begingroup$
    You only seem to be counting the subgroups of order $p^2$. There are others.
    $endgroup$
    – Jeremy Rickard
    Dec 30 '18 at 13:11












  • $begingroup$
    I only wanted those of order p^2, still dont know how to do it though
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:13










  • $begingroup$
    @PedroSantos, I have calculated the result for subgroup of order $p$2$ only
    $endgroup$
    – M. A. SARKAR
    Dec 30 '18 at 13:15










  • $begingroup$
    Yes i know thanks, but im not sure how you did it .
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:16














1












1








1





$begingroup$

The given group is $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$. We will use the formula $$ |(m,n)|=lcm(|m|,|n|),$$ for any $ m in mathbb{Z}_{p^3}$ and $n in mathbb{Z}_{p^2}$, where $|.|$ denote the order .



Since $m,n$ are the elements of $mathbb{Z}_{p^3}$ and $mathbb{Z}_{p^2}$ respectively, we have $ |m| $ divides $ p^3$ and $|n|$ divides $p^2$.



(i) Show that there is only one subgroup of order of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_p oplus mathbb{Z}_p$.



In this step we calculate the number of element $(m,n)$ of order $p$. You easily check that there are $p^2-1$ elements having order $p^2$. All these $p^2-1$ elements along with the identity element $(0,0)$ form a unique subgroup of order $p^2$ that is isomorphic to $mathbb{Z}_p oplus mathbb{Z}_p$.



(ii) Show that there are $p^2+p$ subgroup of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_{p^2}$.



In this step we will count the number of cyclic subgroups of order $p^2$. In other words we have to find all elements $(m,n)$ of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ having order $p^2$. To do this we need to find



$(a)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|m|=p^2 in mathbb{Z}_{p^3}$,



$(b)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|n|=p^2 in mathbb{Z}_{p^2}$



From part $(a)$ and part $(b)$ Try to show that there are total $p^4-p^2$ elements of order $p^2$ and each cyclic group order $p^2$ has $p^2-p$ elements of order $p^2$.



Hence the number of cyclic subgroup of order $p^2$ is equal to $frac{p^4-p^2}{p^2-p} =p^2+p$.



Thus there is a total of $p^2+p+1$ subgroups of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ of order $p^2$.






share|cite|improve this answer











$endgroup$



The given group is $mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$. We will use the formula $$ |(m,n)|=lcm(|m|,|n|),$$ for any $ m in mathbb{Z}_{p^3}$ and $n in mathbb{Z}_{p^2}$, where $|.|$ denote the order .



Since $m,n$ are the elements of $mathbb{Z}_{p^3}$ and $mathbb{Z}_{p^2}$ respectively, we have $ |m| $ divides $ p^3$ and $|n|$ divides $p^2$.



(i) Show that there is only one subgroup of order of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_p oplus mathbb{Z}_p$.



In this step we calculate the number of element $(m,n)$ of order $p$. You easily check that there are $p^2-1$ elements having order $p^2$. All these $p^2-1$ elements along with the identity element $(0,0)$ form a unique subgroup of order $p^2$ that is isomorphic to $mathbb{Z}_p oplus mathbb{Z}_p$.



(ii) Show that there are $p^2+p$ subgroup of $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ isomorphic to $ mathbb{Z}_{p^2}$.



In this step we will count the number of cyclic subgroups of order $p^2$. In other words we have to find all elements $(m,n)$ of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ having order $p^2$. To do this we need to find



$(a)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|m|=p^2 in mathbb{Z}_{p^3}$,



$(b)$ all elements $(m,n) in mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$, such that $|n|=p^2 in mathbb{Z}_{p^2}$



From part $(a)$ and part $(b)$ Try to show that there are total $p^4-p^2$ elements of order $p^2$ and each cyclic group order $p^2$ has $p^2-p$ elements of order $p^2$.



Hence the number of cyclic subgroup of order $p^2$ is equal to $frac{p^4-p^2}{p^2-p} =p^2+p$.



Thus there is a total of $p^2+p+1$ subgroups of the group $ mathbb{Z}_{p^3} oplus mathbb{Z}_{p^2}$ of order $p^2$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 30 '18 at 20:21

























answered Dec 30 '18 at 12:59









M. A. SARKARM. A. SARKAR

2,2621619




2,2621619












  • $begingroup$
    Alright Thanks, But my problem is that i dont know how to prove those statements.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:04










  • $begingroup$
    You only seem to be counting the subgroups of order $p^2$. There are others.
    $endgroup$
    – Jeremy Rickard
    Dec 30 '18 at 13:11












  • $begingroup$
    I only wanted those of order p^2, still dont know how to do it though
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:13










  • $begingroup$
    @PedroSantos, I have calculated the result for subgroup of order $p$2$ only
    $endgroup$
    – M. A. SARKAR
    Dec 30 '18 at 13:15










  • $begingroup$
    Yes i know thanks, but im not sure how you did it .
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:16


















  • $begingroup$
    Alright Thanks, But my problem is that i dont know how to prove those statements.
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:04










  • $begingroup$
    You only seem to be counting the subgroups of order $p^2$. There are others.
    $endgroup$
    – Jeremy Rickard
    Dec 30 '18 at 13:11












  • $begingroup$
    I only wanted those of order p^2, still dont know how to do it though
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:13










  • $begingroup$
    @PedroSantos, I have calculated the result for subgroup of order $p$2$ only
    $endgroup$
    – M. A. SARKAR
    Dec 30 '18 at 13:15










  • $begingroup$
    Yes i know thanks, but im not sure how you did it .
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 13:16
















$begingroup$
Alright Thanks, But my problem is that i dont know how to prove those statements.
$endgroup$
– Pedro Santos
Dec 30 '18 at 13:04




$begingroup$
Alright Thanks, But my problem is that i dont know how to prove those statements.
$endgroup$
– Pedro Santos
Dec 30 '18 at 13:04












$begingroup$
You only seem to be counting the subgroups of order $p^2$. There are others.
$endgroup$
– Jeremy Rickard
Dec 30 '18 at 13:11






$begingroup$
You only seem to be counting the subgroups of order $p^2$. There are others.
$endgroup$
– Jeremy Rickard
Dec 30 '18 at 13:11














$begingroup$
I only wanted those of order p^2, still dont know how to do it though
$endgroup$
– Pedro Santos
Dec 30 '18 at 13:13




$begingroup$
I only wanted those of order p^2, still dont know how to do it though
$endgroup$
– Pedro Santos
Dec 30 '18 at 13:13












$begingroup$
@PedroSantos, I have calculated the result for subgroup of order $p$2$ only
$endgroup$
– M. A. SARKAR
Dec 30 '18 at 13:15




$begingroup$
@PedroSantos, I have calculated the result for subgroup of order $p$2$ only
$endgroup$
– M. A. SARKAR
Dec 30 '18 at 13:15












$begingroup$
Yes i know thanks, but im not sure how you did it .
$endgroup$
– Pedro Santos
Dec 30 '18 at 13:16




$begingroup$
Yes i know thanks, but im not sure how you did it .
$endgroup$
– Pedro Santos
Dec 30 '18 at 13:16











1












$begingroup$

To answer the complete question it is helpful to know that, in a finite abelian group, the number of subgroups of order $n$ is equal to the number of index $n$ for any $n$. This is basically because finite abelian groups $G$ are isomorphic to their dual groups ${rm Hom}(G,{mathbb C}^times)$, and subgroups of the group correspond to quotients of the dual.



So the final answer is $2(1 + (p+1) + (p^2+p+1)) = 2p^2+4p+6$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 17:09






  • 1




    $begingroup$
    That follows from the fact that finite abelian groups are direct sums of cyclic groups.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 18:07
















1












$begingroup$

To answer the complete question it is helpful to know that, in a finite abelian group, the number of subgroups of order $n$ is equal to the number of index $n$ for any $n$. This is basically because finite abelian groups $G$ are isomorphic to their dual groups ${rm Hom}(G,{mathbb C}^times)$, and subgroups of the group correspond to quotients of the dual.



So the final answer is $2(1 + (p+1) + (p^2+p+1)) = 2p^2+4p+6$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 17:09






  • 1




    $begingroup$
    That follows from the fact that finite abelian groups are direct sums of cyclic groups.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 18:07














1












1








1





$begingroup$

To answer the complete question it is helpful to know that, in a finite abelian group, the number of subgroups of order $n$ is equal to the number of index $n$ for any $n$. This is basically because finite abelian groups $G$ are isomorphic to their dual groups ${rm Hom}(G,{mathbb C}^times)$, and subgroups of the group correspond to quotients of the dual.



So the final answer is $2(1 + (p+1) + (p^2+p+1)) = 2p^2+4p+6$.






share|cite|improve this answer









$endgroup$



To answer the complete question it is helpful to know that, in a finite abelian group, the number of subgroups of order $n$ is equal to the number of index $n$ for any $n$. This is basically because finite abelian groups $G$ are isomorphic to their dual groups ${rm Hom}(G,{mathbb C}^times)$, and subgroups of the group correspond to quotients of the dual.



So the final answer is $2(1 + (p+1) + (p^2+p+1)) = 2p^2+4p+6$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 30 '18 at 17:07









Derek HoltDerek Holt

53.8k53571




53.8k53571












  • $begingroup$
    I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 17:09






  • 1




    $begingroup$
    That follows from the fact that finite abelian groups are direct sums of cyclic groups.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 18:07


















  • $begingroup$
    I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
    $endgroup$
    – Pedro Santos
    Dec 30 '18 at 17:09






  • 1




    $begingroup$
    That follows from the fact that finite abelian groups are direct sums of cyclic groups.
    $endgroup$
    – Derek Holt
    Dec 30 '18 at 18:07
















$begingroup$
I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
$endgroup$
– Pedro Santos
Dec 30 '18 at 17:09




$begingroup$
I only knew that finitely generated free modules where isomorphic to their dual, didnt know that a similiar statement worked for finitely generated abelian group and those $Hom$
$endgroup$
– Pedro Santos
Dec 30 '18 at 17:09




1




1




$begingroup$
That follows from the fact that finite abelian groups are direct sums of cyclic groups.
$endgroup$
– Derek Holt
Dec 30 '18 at 18:07




$begingroup$
That follows from the fact that finite abelian groups are direct sums of cyclic groups.
$endgroup$
– Derek Holt
Dec 30 '18 at 18:07


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056775%2ffinding-the-number-of-subgroups-of-mathbbz-p3-oplus-mathbbz-p2%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Bressuire

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more

Cabo Verde

Image
  Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
Read more

Gyllenstierna

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more