If $G'=G$, and suppose $N_G(P)$ is solvable for $Psubseteq G$, show that $ G$ is simple.
$begingroup$
Given a finite nontrivial group $G$, Suppose $N_G(P)$ is solvable for every nontrivial $Psubseteq G$ of prime power order. If $G'=G$, (where $G'$ is the derived subgroup) show that $G$ is simple.
I understand the pieces of information given in the problem but how to put them together to make an argument is challenging to me. I will appreciate any help.
I am thinking that, if I start by assuming $G$ is not simple and that there exists a normal nontrivial subgroup $ Ntriangleleft G$.
And if I can show that the commutator subgroup $G'=1$ then since $G'=G$ I get a contradiction to the nontriviality assumption of the group.
Is this a good way to go? How should I view it?
Thanks
abstract-algebra group-theory finite-groups
edited Dec 4 '18 at 9:39
the_fox
2,89021537
asked Dec 4 '18 at 9:13
CnineCnine
895
$endgroup$
add a comment |
$begingroup$
Given a finite nontrivial group $G$, Suppose $N_G(P)$ is solvable for every nontrivial $Psubseteq G$ of prime power order. If $G'=G$, (where $G'$ is the derived subgroup) show that $G$ is simple.
I understand the pieces of information given in the problem but how to put them together to make an argument is challenging to me. I will appreciate any help.
I am thinking that, if I start by assuming $G$ is not simple and that there exists a normal nontrivial subgroup $ Ntriangleleft G$.
And if I can show that the commutator subgroup $G'=1$ then since $G'=G$ I get a contradiction to the nontriviality assumption of the group.
Is this a good way to go? How should I view it?
Thanks
abstract-algebra group-theory finite-groups
edited Dec 4 '18 at 9:39
the_fox
2,89021537
asked Dec 4 '18 at 9:13
CnineCnine
895
$endgroup$
3
$begingroup$
Choose $P in {rm Syl}_p(N)$ for some prime $p$ dividing $N$. Then by the Frattini Argument $G = NN_G(P)$ and $G/N cong N_G(P)/N_N(P)$. Now use the facts that $N_G(P)$ is solvable and $G'=G$ to deduce that $G=N$.
$endgroup$
– Derek Holt
Dec 4 '18 at 10:05
$begingroup$
@DerekHolt, thank you. but why is $G/N=N_G(P)/N_N(P)$.?
$endgroup$
– Cnine
Dec 5 '18 at 23:08
$begingroup$
That's the second isomorphism theorem
$endgroup$
– Derek Holt
Dec 6 '18 at 8:08
add a comment |
$begingroup$
Given a finite nontrivial group $G$, Suppose $N_G(P)$ is solvable for every nontrivial $Psubseteq G$ of prime power order. If $G'=G$, (where $G'$ is the derived subgroup) show that $G$ is simple.
I understand the pieces of information given in the problem but how to put them together to make an argument is challenging to me. I will appreciate any help.
I am thinking that, if I start by assuming $G$ is not simple and that there exists a normal nontrivial subgroup $ Ntriangleleft G$.
And if I can show that the commutator subgroup $G'=1$ then since $G'=G$ I get a contradiction to the nontriviality assumption of the group.
Is this a good way to go? How should I view it?
Thanks
abstract-algebra group-theory finite-groups
edited Dec 4 '18 at 9:39
the_fox
2,89021537
asked Dec 4 '18 at 9:13
CnineCnine
895
$endgroup$
Given a finite nontrivial group $G$, Suppose $N_G(P)$ is solvable for every nontrivial $Psubseteq G$ of prime power order. If $G'=G$, (where $G'$ is the derived subgroup) show that $G$ is simple.
I understand the pieces of information given in the problem but how to put them together to make an argument is challenging to me. I will appreciate any help.
I am thinking that, if I start by assuming $G$ is not simple and that there exists a normal nontrivial subgroup $ Ntriangleleft G$.
And if I can show that the commutator subgroup $G'=1$ then since $G'=G$ I get a contradiction to the nontriviality assumption of the group.
Is this a good way to go? How should I view it?
Thanks
abstract-algebra group-theory finite-groups
abstract-algebra group-theory finite-groups
edited Dec 4 '18 at 9:39
the_fox
2,89021537
asked Dec 4 '18 at 9:13
CnineCnine
895
edited Dec 4 '18 at 9:39
the_fox
2,89021537
asked Dec 4 '18 at 9:13
CnineCnine
895
edited Dec 4 '18 at 9:39
the_fox
2,89021537
edited Dec 4 '18 at 9:39
the_fox
2,89021537
edited Dec 4 '18 at 9:39
the_fox
2,89021537
2,89021537
asked Dec 4 '18 at 9:13
CnineCnine
895
asked Dec 4 '18 at 9:13
CnineCnine
895
asked Dec 4 '18 at 9:13
CnineCnine
895
895
3
$begingroup$
Choose $P in {rm Syl}_p(N)$ for some prime $p$ dividing $N$. Then by the Frattini Argument $G = NN_G(P)$ and $G/N cong N_G(P)/N_N(P)$. Now use the facts that $N_G(P)$ is solvable and $G'=G$ to deduce that $G=N$.
$endgroup$
– Derek Holt
Dec 4 '18 at 10:05
$begingroup$
@DerekHolt, thank you. but why is $G/N=N_G(P)/N_N(P)$.?
$endgroup$
– Cnine
Dec 5 '18 at 23:08
$begingroup$
That's the second isomorphism theorem
$endgroup$
– Derek Holt
Dec 6 '18 at 8:08
add a comment |
3
$begingroup$
Choose $P in {rm Syl}_p(N)$ for some prime $p$ dividing $N$. Then by the Frattini Argument $G = NN_G(P)$ and $G/N cong N_G(P)/N_N(P)$. Now use the facts that $N_G(P)$ is solvable and $G'=G$ to deduce that $G=N$.
$endgroup$
– Derek Holt
Dec 4 '18 at 10:05
$begingroup$
@DerekHolt, thank you. but why is $G/N=N_G(P)/N_N(P)$.?
$endgroup$
– Cnine
Dec 5 '18 at 23:08
$begingroup$
That's the second isomorphism theorem
$endgroup$
– Derek Holt
Dec 6 '18 at 8:08
3
3
$begingroup$
Choose $P in {rm Syl}_p(N)$ for some prime $p$ dividing $N$. Then by the Frattini Argument $G = NN_G(P)$ and $G/N cong N_G(P)/N_N(P)$. Now use the facts that $N_G(P)$ is solvable and $G'=G$ to deduce that $G=N$.
$endgroup$
– Derek Holt
Dec 4 '18 at 10:05
$begingroup$
Choose $P in {rm Syl}_p(N)$ for some prime $p$ dividing $N$. Then by the Frattini Argument $G = NN_G(P)$ and $G/N cong N_G(P)/N_N(P)$. Now use the facts that $N_G(P)$ is solvable and $G'=G$ to deduce that $G=N$.
$endgroup$
– Derek Holt
Dec 4 '18 at 10:05
$begingroup$
@DerekHolt, thank you. but why is $G/N=N_G(P)/N_N(P)$.?
$endgroup$
– Cnine
Dec 5 '18 at 23:08
$begingroup$
@DerekHolt, thank you. but why is $G/N=N_G(P)/N_N(P)$.?
$endgroup$
– Cnine
Dec 5 '18 at 23:08
$begingroup$
That's the second isomorphism theorem
$endgroup$
– Derek Holt
Dec 6 '18 at 8:08
$begingroup$
That's the second isomorphism theorem
$endgroup$
– Derek Holt
Dec 6 '18 at 8:08
add a comment |
0
active
oldest
votes
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3025326%2fif-g-g-and-suppose-n-gp-is-solvable-for-p-subseteq-g-show-that-g-i%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3025326%2fif-g-g-and-suppose-n-gp-is-solvable-for-p-subseteq-g-show-that-g-i%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
3
$begingroup$
Choose $P in {rm Syl}_p(N)$ for some prime $p$ dividing $N$. Then by the Frattini Argument $G = NN_G(P)$ and $G/N cong N_G(P)/N_N(P)$. Now use the facts that $N_G(P)$ is solvable and $G'=G$ to deduce that $G=N$.
$endgroup$
– Derek Holt
Dec 4 '18 at 10:05
$begingroup$
@DerekHolt, thank you. but why is $G/N=N_G(P)/N_N(P)$.?
$endgroup$
– Cnine
Dec 5 '18 at 23:08
$begingroup$
That's the second isomorphism theorem
$endgroup$
– Derek Holt
Dec 6 '18 at 8:08