Determining whether a group is cyclic by finding its subgroups
$begingroup$
In my text book we are given the question: 'Is $mathbb{Z}^*_{15}$ cyclic?'
And in the answers we are given: 'We find that in this group, the subgroups generated by the elements are {1}, {1, 4}, {1, 11}, {1, 14}, {1, 2, 4, 8}, and {1, 7, 4, 13}. Since none of these is equal to the whole group, we deduce that it is not cyclic.'
I understand that every subgroup must contain the identity {1} and that $4^2 mod15$ gives the identity giving the group {1, 4}. We can also do the same thing with {1, 11}, {1, 14} but i don't understand how they got {1, 2, 4, 8} and {1, 7, 4, 13}.
group-theory
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
$endgroup$
add a comment |
$begingroup$
In my text book we are given the question: 'Is $mathbb{Z}^*_{15}$ cyclic?'
And in the answers we are given: 'We find that in this group, the subgroups generated by the elements are {1}, {1, 4}, {1, 11}, {1, 14}, {1, 2, 4, 8}, and {1, 7, 4, 13}. Since none of these is equal to the whole group, we deduce that it is not cyclic.'
I understand that every subgroup must contain the identity {1} and that $4^2 mod15$ gives the identity giving the group {1, 4}. We can also do the same thing with {1, 11}, {1, 14} but i don't understand how they got {1, 2, 4, 8} and {1, 7, 4, 13}.
group-theory
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
$endgroup$
add a comment |
$begingroup$
In my text book we are given the question: 'Is $mathbb{Z}^*_{15}$ cyclic?'
And in the answers we are given: 'We find that in this group, the subgroups generated by the elements are {1}, {1, 4}, {1, 11}, {1, 14}, {1, 2, 4, 8}, and {1, 7, 4, 13}. Since none of these is equal to the whole group, we deduce that it is not cyclic.'
I understand that every subgroup must contain the identity {1} and that $4^2 mod15$ gives the identity giving the group {1, 4}. We can also do the same thing with {1, 11}, {1, 14} but i don't understand how they got {1, 2, 4, 8} and {1, 7, 4, 13}.
group-theory
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
$endgroup$
In my text book we are given the question: 'Is $mathbb{Z}^*_{15}$ cyclic?'
And in the answers we are given: 'We find that in this group, the subgroups generated by the elements are {1}, {1, 4}, {1, 11}, {1, 14}, {1, 2, 4, 8}, and {1, 7, 4, 13}. Since none of these is equal to the whole group, we deduce that it is not cyclic.'
I understand that every subgroup must contain the identity {1} and that $4^2 mod15$ gives the identity giving the group {1, 4}. We can also do the same thing with {1, 11}, {1, 14} but i don't understand how they got {1, 2, 4, 8} and {1, 7, 4, 13}.
group-theory
group-theory
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
asked Jan 9 at 14:55
Dennis HoustonDennis Houston
1
1
add a comment |
add a comment |
1 Answer
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$begingroup$
${1,2,4,8}$ simply is the subgroup generated by $2$ because $2^2=4$ mod $15$, $2^3=8$ mod $15$ and then $2^4 = 16 = 1$ mod $15$. Similarly, ${1, 7, 4, 13}$ is the subgroup generated by $7$.
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
$endgroup$
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
1
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
add a comment |
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$begingroup$
${1,2,4,8}$ simply is the subgroup generated by $2$ because $2^2=4$ mod $15$, $2^3=8$ mod $15$ and then $2^4 = 16 = 1$ mod $15$. Similarly, ${1, 7, 4, 13}$ is the subgroup generated by $7$.
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
$endgroup$
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
1
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
add a comment |
$begingroup$
${1,2,4,8}$ simply is the subgroup generated by $2$ because $2^2=4$ mod $15$, $2^3=8$ mod $15$ and then $2^4 = 16 = 1$ mod $15$. Similarly, ${1, 7, 4, 13}$ is the subgroup generated by $7$.
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
$endgroup$
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
1
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
add a comment |
$begingroup$
${1,2,4,8}$ simply is the subgroup generated by $2$ because $2^2=4$ mod $15$, $2^3=8$ mod $15$ and then $2^4 = 16 = 1$ mod $15$. Similarly, ${1, 7, 4, 13}$ is the subgroup generated by $7$.
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
$endgroup$
${1,2,4,8}$ simply is the subgroup generated by $2$ because $2^2=4$ mod $15$, $2^3=8$ mod $15$ and then $2^4 = 16 = 1$ mod $15$. Similarly, ${1, 7, 4, 13}$ is the subgroup generated by $7$.
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
answered Jan 9 at 14:57
A. BailleulA. Bailleul
1587
1587
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
1
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
add a comment |
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
1
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
How come they didnt include the subgroup generated by other numbers in the group e.g. 3?
$endgroup$
– Dennis Houston
Jan 9 at 14:59
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
$3$ is not in the group.
$endgroup$
– Derek Holt
Jan 9 at 15:00
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
$begingroup$
Surely the group is the integers 1 - 14 inclusive? mod 15
$endgroup$
– Dennis Houston
Jan 9 at 15:02
1
1
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
No, the group consists in the $varphi (15)=8$ numbers (or residues, actually) relatively prime to $15$. For instance, $3$ and $15$ are not coprime.
$endgroup$
– Chris Custer
Jan 9 at 15:11
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
$begingroup$
Oh thank you, that explains a lot
$endgroup$
– Dennis Houston
Jan 9 at 15:14
add a comment |
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