Number of generators of subgroup Announcing the arrival of Valued Associate #679: Cesar...
What did Turing mean when saying that "machines cannot give rise to surprises" is due to a fallacy?
Any stored/leased 737s that could substitute for grounded MAXs?
An isoperimetric-type inequality inside a cube
Did pre-Columbian Americans know the spherical shape of the Earth?
The Nth Gryphon Number
In musical terms, what properties are varied by the human voice to produce different words / syllables?
Noise in Eigenvalues plot
Who said what about *meanings*?
Is this Kuo-toa homebrew race balanced?
Did any compiler fully use 80-bit floating point?
Can gravitational waves pass through a black hole?
Can the Haste spell grant both a Beast Master ranger and their animal companion extra attacks?
Is the time—manner—place ordering of adverbials an oversimplification?
Does the main washing effect of soap come from foam?
Pointing to problems without suggesting solutions
How to make triangles with rounded sides and corners? (squircle with 3 sides)
systemd and copy (/bin/cp): no such file or directory
By what mechanism was the 2017 UK General Election called?
How does the body cool itself in a stillsuit?
What does 丫 mean? 丫是什么意思？
Random body shuffle every night—can we still function?
As a dual citizen, my US passport will expire one day after traveling to the US. Will this work?
Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?
What is the proper term for etching or digging of wall to hide conduit of cables
Number of generators of subgroup
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Torsion subgroupOn the minimal number of generators of a finite groupBound number of generators of a subgroup of a nilpotent group?Minimal number of generators for a finitely generated abelian $p$-groupA question on finitely generated Abelian groups with a minimal number of generatorsFactoring an Abelian groupThe number of internal direct summands of a finitely generated abelian groupFree group generated by two generators is isomorphic to product of two infinite cyclic groupsAlternative proof of the Fundamental Theorem of Abelian Groups??Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups
$begingroup$
I am trying to prove the following.
let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?
Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbb{Z}$ and $H=2mathbb{Z}$.
I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.
group-theory abelian-groups
asked 5 hours ago
CharlesCharles
582420
$endgroup$
add a comment |
$begingroup$
I am trying to prove the following.
let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?
Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbb{Z}$ and $H=2mathbb{Z}$.
I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.
group-theory abelian-groups
asked 5 hours ago
CharlesCharles
582420
$endgroup$
$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
4 hours ago
2
$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
4 hours ago
$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
4 hours ago
add a comment |
$begingroup$
I am trying to prove the following.
let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?
Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbb{Z}$ and $H=2mathbb{Z}$.
I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.
group-theory abelian-groups
asked 5 hours ago
CharlesCharles
582420
$endgroup$
I am trying to prove the following.
let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H oplus K$. Is it true that the minimal number of generators of H is strictly smaller than the minimal number of generators of $G$?
Clearly if G can not be written as a direct summand of $H$ then this is not true, just consider $G= mathbb{Z}$ and $H=2mathbb{Z}$.
I would like to prove it because I believe it can provide a simpler proof for the characterization of finitely generated abelian groups.
group-theory abelian-groups
group-theory abelian-groups
asked 5 hours ago
CharlesCharles
582420
asked 5 hours ago
CharlesCharles
582420
edited 4 hours ago
Charles
asked 5 hours ago
CharlesCharles
582420
asked 5 hours ago
CharlesCharles
582420
asked 5 hours ago
CharlesCharles
582420
582420
$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
4 hours ago
2
$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
4 hours ago
$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
4 hours ago
add a comment |
$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
4 hours ago
2
$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
4 hours ago
$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
4 hours ago
$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
4 hours ago
$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
4 hours ago
2
2
$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
4 hours ago
$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
4 hours ago
$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
4 hours ago
$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
4 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, it is not true. Consider $mathbb{Z}_2oplusmathbb{Z}_3$. This has a generator $(1,1)$. Note that
$$0oplusmathbb{Z}_3<mathbb{Z}_2oplusmathbb{Z}_3 ,$$
and
$$(mathbb{Z}_2oplus 0)oplus(0oplusmathbb{Z_3})=mathbb{Z}_2oplusmathbb{Z}_3.$$
However, $0oplusmathbb{Z}_3$ is generated by $(0,1).$
answered 4 hours ago
MelodyMelody
1,41212
$endgroup$
add a comment |
Your Answer
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%2f3196206%2fnumber-of-generators-of-subgroup%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, it is not true. Consider $mathbb{Z}_2oplusmathbb{Z}_3$. This has a generator $(1,1)$. Note that
$$0oplusmathbb{Z}_3<mathbb{Z}_2oplusmathbb{Z}_3 ,$$
and
$$(mathbb{Z}_2oplus 0)oplus(0oplusmathbb{Z_3})=mathbb{Z}_2oplusmathbb{Z}_3.$$
However, $0oplusmathbb{Z}_3$ is generated by $(0,1).$
answered 4 hours ago
MelodyMelody
1,41212
$endgroup$
add a comment |
$begingroup$
No, it is not true. Consider $mathbb{Z}_2oplusmathbb{Z}_3$. This has a generator $(1,1)$. Note that
$$0oplusmathbb{Z}_3<mathbb{Z}_2oplusmathbb{Z}_3 ,$$
and
$$(mathbb{Z}_2oplus 0)oplus(0oplusmathbb{Z_3})=mathbb{Z}_2oplusmathbb{Z}_3.$$
However, $0oplusmathbb{Z}_3$ is generated by $(0,1).$
answered 4 hours ago
MelodyMelody
1,41212
$endgroup$
add a comment |
$begingroup$
No, it is not true. Consider $mathbb{Z}_2oplusmathbb{Z}_3$. This has a generator $(1,1)$. Note that
$$0oplusmathbb{Z}_3<mathbb{Z}_2oplusmathbb{Z}_3 ,$$
and
$$(mathbb{Z}_2oplus 0)oplus(0oplusmathbb{Z_3})=mathbb{Z}_2oplusmathbb{Z}_3.$$
However, $0oplusmathbb{Z}_3$ is generated by $(0,1).$
answered 4 hours ago
MelodyMelody
1,41212
$endgroup$
No, it is not true. Consider $mathbb{Z}_2oplusmathbb{Z}_3$. This has a generator $(1,1)$. Note that
$$0oplusmathbb{Z}_3<mathbb{Z}_2oplusmathbb{Z}_3 ,$$
and
$$(mathbb{Z}_2oplus 0)oplus(0oplusmathbb{Z_3})=mathbb{Z}_2oplusmathbb{Z}_3.$$
However, $0oplusmathbb{Z}_3$ is generated by $(0,1).$
answered 4 hours ago
MelodyMelody
1,41212
answered 4 hours ago
MelodyMelody
1,41212
answered 4 hours ago
MelodyMelody
1,41212
answered 4 hours ago
MelodyMelody
1,41212
1,41212
add a comment |
add a comment |
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%2f3196206%2fnumber-of-generators-of-subgroup%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
$begingroup$
$mathbb Zbig / 2mathbb Z oplus mathbb Zbig / 3mathbb Z $ is cyclic.
$endgroup$
– lulu
4 hours ago
2
$begingroup$
Worth noting: "number of generators" is not well defined. I'm guessing you mean "minimal number of generators", but you should say so,
$endgroup$
– lulu
4 hours ago
$begingroup$
Thank you for pointing that out. I will edit to correct it.
$endgroup$
– Charles
4 hours ago