Kernel is a subgroup of the normalizer of p-subgroup
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Im trying to understand the proof in the book Abstract Algebra by Dummit.
Prove: G of order 12 has a sylow 3 subgroup or isomorphic to A4
.....
.
.
Since distinct Sylow 3-subgroups intersect in the identity and each contains two elements of order 3, $G$ contains $2•4=8$ elements of order $3$. Since $|G:N_G(P)| = n_3 = 4, N_G(P) = P$. Now $G$ acts by conjugation in its four Sylow 3-subgroups, so this action affords a permutation representation $phi : G to S_4$. The kernel $K$ of this action is the subgroup of $G$ which normalizes all Sylow $3$-subgroups of $G$. In particular, $K leq N_G(P) = P$. Since $P$ is not normal in $G$ by assumption, $K=1$
Why is it that $K leq N_G(P)?$
abstract-algebra
asked Nov 18 at 10:01
Heisenberg
13
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Im trying to understand the proof in the book Abstract Algebra by Dummit.
Prove: G of order 12 has a sylow 3 subgroup or isomorphic to A4
.....
.
.
Since distinct Sylow 3-subgroups intersect in the identity and each contains two elements of order 3, $G$ contains $2•4=8$ elements of order $3$. Since $|G:N_G(P)| = n_3 = 4, N_G(P) = P$. Now $G$ acts by conjugation in its four Sylow 3-subgroups, so this action affords a permutation representation $phi : G to S_4$. The kernel $K$ of this action is the subgroup of $G$ which normalizes all Sylow $3$-subgroups of $G$. In particular, $K leq N_G(P) = P$. Since $P$ is not normal in $G$ by assumption, $K=1$
Why is it that $K leq N_G(P)?$
abstract-algebra
asked Nov 18 at 10:01
Heisenberg
13
Because the elements in $;N_G(P);$ are precisely all those that normalize $;P;$, so if $;K;$ normalize $;P;$ ...
– DonAntonio
Nov 18 at 10:52
add a comment |
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up vote
0
down vote
favorite
Im trying to understand the proof in the book Abstract Algebra by Dummit.
Prove: G of order 12 has a sylow 3 subgroup or isomorphic to A4
.....
.
.
Since distinct Sylow 3-subgroups intersect in the identity and each contains two elements of order 3, $G$ contains $2•4=8$ elements of order $3$. Since $|G:N_G(P)| = n_3 = 4, N_G(P) = P$. Now $G$ acts by conjugation in its four Sylow 3-subgroups, so this action affords a permutation representation $phi : G to S_4$. The kernel $K$ of this action is the subgroup of $G$ which normalizes all Sylow $3$-subgroups of $G$. In particular, $K leq N_G(P) = P$. Since $P$ is not normal in $G$ by assumption, $K=1$
Why is it that $K leq N_G(P)?$
abstract-algebra
asked Nov 18 at 10:01
Heisenberg
13
Im trying to understand the proof in the book Abstract Algebra by Dummit.
Prove: G of order 12 has a sylow 3 subgroup or isomorphic to A4
.....
.
.
Since distinct Sylow 3-subgroups intersect in the identity and each contains two elements of order 3, $G$ contains $2•4=8$ elements of order $3$. Since $|G:N_G(P)| = n_3 = 4, N_G(P) = P$. Now $G$ acts by conjugation in its four Sylow 3-subgroups, so this action affords a permutation representation $phi : G to S_4$. The kernel $K$ of this action is the subgroup of $G$ which normalizes all Sylow $3$-subgroups of $G$. In particular, $K leq N_G(P) = P$. Since $P$ is not normal in $G$ by assumption, $K=1$
Why is it that $K leq N_G(P)?$
abstract-algebra
abstract-algebra
asked Nov 18 at 10:01
Heisenberg
13
asked Nov 18 at 10:01
Heisenberg
13
asked Nov 18 at 10:01
Heisenberg
13
asked Nov 18 at 10:01
Heisenberg
13
asked Nov 18 at 10:01
Heisenberg
13
13
Because the elements in $;N_G(P);$ are precisely all those that normalize $;P;$, so if $;K;$ normalize $;P;$ ...
– DonAntonio
Nov 18 at 10:52
add a comment |
Because the elements in $;N_G(P);$ are precisely all those that normalize $;P;$, so if $;K;$ normalize $;P;$ ...
– DonAntonio
Nov 18 at 10:52
Because the elements in $;N_G(P);$ are precisely all those that normalize $;P;$, so if $;K;$ normalize $;P;$ ...
– DonAntonio
Nov 18 at 10:52
Because the elements in $;N_G(P);$ are precisely all those that normalize $;P;$, so if $;K;$ normalize $;P;$ ...
– DonAntonio
Nov 18 at 10:52
add a comment |
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Because the elements in $;N_G(P);$ are precisely all those that normalize $;P;$, so if $;K;$ normalize $;P;$ ...
– DonAntonio
Nov 18 at 10:52