Normal Subgroups and Quotient Groups help
$begingroup$
Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$.
(a) Prove that if $xH in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$.
(b) Suppose that there is no positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = ∞$.
(c) Suppose that there exists a positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = d$, where $d$ is the least positive integer for which $x^d ∈ H$.
normal-subgroups quotient-group
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
$endgroup$
add a comment |
$begingroup$
Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$.
(a) Prove that if $xH in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$.
(b) Suppose that there is no positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = ∞$.
(c) Suppose that there exists a positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = d$, where $d$ is the least positive integer for which $x^d ∈ H$.
normal-subgroups quotient-group
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
$endgroup$
$begingroup$
Welcome to Maths SX! What did you try and where are you stuck?
$endgroup$
– Bernard
Jan 6 at 16:51
$begingroup$
I edited your post to make $LaTeX$ work, and to bring it into accord with typical mathematical conventions. Hope I got it right! Cheers!
$endgroup$
– Robert Lewis
Jan 6 at 17:37
$begingroup$
Thank you! That is looking good. I have proven part (a) by doing xH ⋆ xH ⋆ ... ⋆ xH (m times) which gives us ( x * x * ... * x) (also m times).
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
But I do not know how to prove part (b) and (c) yet. Anyone could give me a hand?
$endgroup$
– Marwan Helali
Jan 6 at 19:36
add a comment |
$begingroup$
Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$.
(a) Prove that if $xH in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$.
(b) Suppose that there is no positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = ∞$.
(c) Suppose that there exists a positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = d$, where $d$ is the least positive integer for which $x^d ∈ H$.
normal-subgroups quotient-group
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
$endgroup$
Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$.
(a) Prove that if $xH in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$.
(b) Suppose that there is no positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = ∞$.
(c) Suppose that there exists a positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = d$, where $d$ is the least positive integer for which $x^d ∈ H$.
normal-subgroups quotient-group
normal-subgroups quotient-group
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
edited Jan 6 at 17:35
Robert Lewis
48.3k23167
48.3k23167
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
asked Jan 6 at 16:34
Marwan HelaliMarwan Helali
161
161
$begingroup$
Welcome to Maths SX! What did you try and where are you stuck?
$endgroup$
– Bernard
Jan 6 at 16:51
$begingroup$
I edited your post to make $LaTeX$ work, and to bring it into accord with typical mathematical conventions. Hope I got it right! Cheers!
$endgroup$
– Robert Lewis
Jan 6 at 17:37
$begingroup$
Thank you! That is looking good. I have proven part (a) by doing xH ⋆ xH ⋆ ... ⋆ xH (m times) which gives us ( x * x * ... * x) (also m times).
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
But I do not know how to prove part (b) and (c) yet. Anyone could give me a hand?
$endgroup$
– Marwan Helali
Jan 6 at 19:36
add a comment |
$begingroup$
Welcome to Maths SX! What did you try and where are you stuck?
$endgroup$
– Bernard
Jan 6 at 16:51
$begingroup$
I edited your post to make $LaTeX$ work, and to bring it into accord with typical mathematical conventions. Hope I got it right! Cheers!
$endgroup$
– Robert Lewis
Jan 6 at 17:37
$begingroup$
Thank you! That is looking good. I have proven part (a) by doing xH ⋆ xH ⋆ ... ⋆ xH (m times) which gives us ( x * x * ... * x) (also m times).
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
But I do not know how to prove part (b) and (c) yet. Anyone could give me a hand?
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
Welcome to Maths SX! What did you try and where are you stuck?
$endgroup$
– Bernard
Jan 6 at 16:51
$begingroup$
Welcome to Maths SX! What did you try and where are you stuck?
$endgroup$
– Bernard
Jan 6 at 16:51
$begingroup$
I edited your post to make $LaTeX$ work, and to bring it into accord with typical mathematical conventions. Hope I got it right! Cheers!
$endgroup$
– Robert Lewis
Jan 6 at 17:37
$begingroup$
I edited your post to make $LaTeX$ work, and to bring it into accord with typical mathematical conventions. Hope I got it right! Cheers!
$endgroup$
– Robert Lewis
Jan 6 at 17:37
$begingroup$
Thank you! That is looking good. I have proven part (a) by doing xH ⋆ xH ⋆ ... ⋆ xH (m times) which gives us ( x * x * ... * x) (also m times).
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
Thank you! That is looking good. I have proven part (a) by doing xH ⋆ xH ⋆ ... ⋆ xH (m times) which gives us ( x * x * ... * x) (also m times).
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
But I do not know how to prove part (b) and (c) yet. Anyone could give me a hand?
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
But I do not know how to prove part (b) and (c) yet. Anyone could give me a hand?
$endgroup$
– Marwan Helali
Jan 6 at 19:36
add a comment |
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$begingroup$
Welcome to Maths SX! What did you try and where are you stuck?
$endgroup$
– Bernard
Jan 6 at 16:51
$begingroup$
I edited your post to make $LaTeX$ work, and to bring it into accord with typical mathematical conventions. Hope I got it right! Cheers!
$endgroup$
– Robert Lewis
Jan 6 at 17:37
$begingroup$
Thank you! That is looking good. I have proven part (a) by doing xH ⋆ xH ⋆ ... ⋆ xH (m times) which gives us ( x * x * ... * x) (also m times).
$endgroup$
– Marwan Helali
Jan 6 at 19:36
$begingroup$
But I do not know how to prove part (b) and (c) yet. Anyone could give me a hand?
$endgroup$
– Marwan Helali
Jan 6 at 19:36