Quotient of finite, nilpotent group by Frattini subgroup is isomorphic to product of quotients of Sylow...
0
$begingroup$
Let $G$ be a finite nilpotent group.
We know that $G=G_{p_1}times G_{p_2}times cdots times G_{p_r}$ where $G_{p_i}in Syl_{p_i}(G)$, $i=1,dots,r$.
Is the following equation right? And why?
$G/Phi(G)=G_{p_1}/Phi(G_{p_1})timescdotstimes G_{p_r}/Phi(G_{p_r})$,
where $Phi(G)$ is the Frattini subgroup of $G$.
finite-groups group-isomorphism sylow-theory direct-product nilpotent-groups
edited Dec 22 '18 at 4:43
Shaun
9,380113684
asked Jul 26 '18 at 9:38
QinQin
200117
$endgroup$
add a comment |
0
$begingroup$
Let $G$ be a finite nilpotent group.
We know that $G=G_{p_1}times G_{p_2}times cdots times G_{p_r}$ where $G_{p_i}in Syl_{p_i}(G)$, $i=1,dots,r$.
Is the following equation right? And why?
$G/Phi(G)=G_{p_1}/Phi(G_{p_1})timescdotstimes G_{p_r}/Phi(G_{p_r})$,
where $Phi(G)$ is the Frattini subgroup of $G$.
finite-groups group-isomorphism sylow-theory direct-product nilpotent-groups
edited Dec 22 '18 at 4:43
Shaun
9,380113684
asked Jul 26 '18 at 9:38
QinQin
200117
$endgroup$
1
$begingroup$
What's $Phi(G)$? Frattini subgroup?
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:42
$begingroup$
@Lord Shark the Unknown Yes!
$endgroup$
– Qin
Jul 26 '18 at 9:43
1
$begingroup$
The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:49
2
$begingroup$
Yes: An "n" is missing.
$endgroup$
– Christian Blatter
Jul 26 '18 at 10:08
add a comment |
0
0
0
$begingroup$
Let $G$ be a finite nilpotent group.
We know that $G=G_{p_1}times G_{p_2}times cdots times G_{p_r}$ where $G_{p_i}in Syl_{p_i}(G)$, $i=1,dots,r$.
Is the following equation right? And why?
$G/Phi(G)=G_{p_1}/Phi(G_{p_1})timescdotstimes G_{p_r}/Phi(G_{p_r})$,
where $Phi(G)$ is the Frattini subgroup of $G$.
finite-groups group-isomorphism sylow-theory direct-product nilpotent-groups
edited Dec 22 '18 at 4:43
Shaun
9,380113684
asked Jul 26 '18 at 9:38
QinQin
200117
$endgroup$
Let $G$ be a finite nilpotent group.
We know that $G=G_{p_1}times G_{p_2}times cdots times G_{p_r}$ where $G_{p_i}in Syl_{p_i}(G)$, $i=1,dots,r$.
Is the following equation right? And why?
$G/Phi(G)=G_{p_1}/Phi(G_{p_1})timescdotstimes G_{p_r}/Phi(G_{p_r})$,
where $Phi(G)$ is the Frattini subgroup of $G$.
finite-groups group-isomorphism sylow-theory direct-product nilpotent-groups
finite-groups group-isomorphism sylow-theory direct-product nilpotent-groups
edited Dec 22 '18 at 4:43
Shaun
9,380113684
asked Jul 26 '18 at 9:38
QinQin
200117
edited Dec 22 '18 at 4:43
Shaun
9,380113684
asked Jul 26 '18 at 9:38
QinQin
200117
edited Dec 22 '18 at 4:43
Shaun
9,380113684
edited Dec 22 '18 at 4:43
Shaun
9,380113684
edited Dec 22 '18 at 4:43
Shaun
9,380113684
9,380113684
asked Jul 26 '18 at 9:38
QinQin
200117
asked Jul 26 '18 at 9:38
QinQin
200117
asked Jul 26 '18 at 9:38
QinQin
200117
200117
1
$begingroup$
What's $Phi(G)$? Frattini subgroup?
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:42
$begingroup$
@Lord Shark the Unknown Yes!
$endgroup$
– Qin
Jul 26 '18 at 9:43
1
$begingroup$
The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:49
2
$begingroup$
Yes: An "n" is missing.
$endgroup$
– Christian Blatter
Jul 26 '18 at 10:08
add a comment |
1
$begingroup$
What's $Phi(G)$? Frattini subgroup?
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:42
$begingroup$
@Lord Shark the Unknown Yes!
$endgroup$
– Qin
Jul 26 '18 at 9:43
1
$begingroup$
The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:49
2
$begingroup$
Yes: An "n" is missing.
$endgroup$
– Christian Blatter
Jul 26 '18 at 10:08
1
1
$begingroup$
What's $Phi(G)$? Frattini subgroup?
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:42
$begingroup$
What's $Phi(G)$? Frattini subgroup?
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:42
$begingroup$
@Lord Shark the Unknown Yes!
$endgroup$
– Qin
Jul 26 '18 at 9:43
$begingroup$
@Lord Shark the Unknown Yes!
$endgroup$
– Qin
Jul 26 '18 at 9:43
1
1
$begingroup$
The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:49
$begingroup$
The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:49
2
2
$begingroup$
Yes: An "n" is missing.
$endgroup$
– Christian Blatter
Jul 26 '18 at 10:08
$begingroup$
Yes: An "n" is missing.
$endgroup$
– Christian Blatter
Jul 26 '18 at 10:08
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
});
}
});
draft saved
draft discarded
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%2f2863253%2fquotient-of-finite-nilpotent-group-by-frattini-subgroup-is-isomorphic-to-produc%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
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
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%2f2863253%2fquotient-of-finite-nilpotent-group-by-frattini-subgroup-is-isomorphic-to-produc%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
1
$begingroup$
What's $Phi(G)$? Frattini subgroup?
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:42
$begingroup$
@Lord Shark the Unknown Yes!
$endgroup$
– Qin
Jul 26 '18 at 9:43
1
$begingroup$
The Frattini subgroup satisfies $Phi(H_1times H_2)=Phi(H_1)timesPhi(H_2)$ so then $(H_1times H_2)/Phi(H_1times H_2)=H_1/Phi(H_1)times H_2/Phi(H_2)$ etc.
$endgroup$
– Lord Shark the Unknown
Jul 26 '18 at 9:49
2
$begingroup$
Yes: An "n" is missing.
$endgroup$
– Christian Blatter
Jul 26 '18 at 10:08