User kabenyuk

10 votes

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Suppose $G=\left\langle x, y, t\mid x^7=y^7=t^3=1, txt^{-1}=x^2, tyt^{-1}=y\right\rangle$. Show that $y\in Z(G)$.

answered Dec 4, 2022 at 5:14

9 votes

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Compact simple group which is not a Lie group

answered Sep 23, 2022 at 13:37

8 votes

Classification of subgroups of $(\mathbb Q,+)$ that are not finitely generated.

answered Aug 1, 2022 at 16:54

7 votes

Examples of a number $d$ that is a divisor of the order of a group $G$, but there is no quotient group $G/N$ whose order is $d$.

answered Mar 29, 2021 at 7:09

7 votes

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A finite group of order $mn$ with $m,n$ relatively prime, together with subgroups of orders $m, n$.

answered Jan 1 at 12:45

7 votes

Example of a non normal subgroup of finite index in an infinite group

answered Nov 17, 2022 at 10:21

6 votes

Find the minimum number of edges in a graph with $3n+1$ vertices if ...

answered Jul 9, 2021 at 8:45

6 votes

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Showing there is a node in the graph with one and only one edge

answered Dec 4, 2021 at 14:59

6 votes

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Does every $3$-regular bipartite graph have a $4$-cycle?

answered Jan 19 at 11:34

6 votes

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Classification of groups such that the converse to Lagrange holds

answered May 29 at 14:57

6 votes

Let G be a non abelian group with a square-free order. Prove that there are two elements $a,b\in G$ such that $ab\neq ba$ and $ord(a)=ord(b)$.

answered Nov 4, 2022 at 11:14

5 votes

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Prove that if $G$ is a finite simple group containing a subgroup $H$ of order $27$, then $|G : H| \ge 9.$

answered Mar 10, 2021 at 17:23

5 votes

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A group such that all its subgroups are finitely generated

answered Mar 12, 2021 at 10:21

5 votes

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Planar graph K3,3

answered Jan 28 at 16:37

5 votes

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Find the value of an expression related to the roots of a quadratic polynomial

answered Sep 11, 2022 at 15:12

5 votes

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How to find subgroups of a product group containing the diagonal?

answered Jul 17, 2022 at 16:14

5 votes

How many planar graphs of vertex degrees all ≥4 are there?, extended (by request of G.M.)

answered Jul 30, 2022 at 17:48

5 votes

An example of a non-diagonalisable matrix in $\mathrm{SL}(n, \mathbb{Z})$ whose eigenvalues don't all have absolute value $1$

answered Jun 5, 2022 at 11:22

4 votes

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If $|G|=n>1$, then $|{\rm Aut}(G)|\le\prod_{i=0}^k(n-2^i)$ for $k=[\log_2(n-1)]$.

answered Mar 13, 2021 at 18:42

4 votes

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Is the intersection of a descending chain of finitely generated groups finitely generated

answered Mar 6, 2021 at 3:04

4 votes

Number of elements of order coprime to $p$ in a finite group

answered Apr 30, 2021 at 16:25

4 votes

Prove that : $f(n) \le \frac{1}{4}(n-1)^2+1 $

answered Jul 20, 2021 at 6:58

4 votes

If $a_n>0$ for all $n$, and $\lim_{n\to\infty}a_na_{n+1}=A$, $\lim_{n\to\infty}a_na_{n+2}=B$, $\lim_{n \to \infty}a_na_{n+3}=C$, dis/prove $A=B=C$

answered Aug 20, 2021 at 10:45

4 votes

Accepted

If $0\leq X \leq \text{Id}$ and $0\leq A$, then $XAX \leq A$?

answered Aug 20, 2021 at 13:25

4 votes

Let $x$ be $(1\dots n)\in S_n$ show that there exists $y\in x^{S_n}$ such that $x^{S_n} = x^{A_n}\sqcup y^{A_n}$.

answered Nov 21, 2021 at 5:21

4 votes

Accepted

Confusion in understanding of Steinitz Exchange Lemma by an example

answered Oct 14, 2021 at 11:31

4 votes

Do these very VERY weak axioms guarantee a group? (Every element has left identity $e_L$ or right identity $e_R$, with same-sided inverse)

answered Sep 17, 2021 at 16:33

4 votes

Let $G$ be generated by $\begin{pmatrix}1&p\\ 0&1\end{pmatrix},\begin{pmatrix}1&0\\ p&1\end{pmatrix}$ for prime $p$. Is $G$ is solvable? A proof.

answered Jan 5 at 10:27

4 votes

Proof that a graph with at most $n+2$ edges is planar

answered Feb 12 at 7:03

4 votes

Accepted

Number of bipartite graphs given two sets

answered Nov 10 at 14:35

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358 Answers

Suppose $G=\left\langle x, y, t\mid x^7=y^7=t^3=1, txt^{-1}=x^2, tyt^{-1}=y\right\rangle$. Show that $y\in Z(G)$.

Compact simple group which is not a Lie group

Classification of subgroups of $(\mathbb Q,+)$ that are not finitely generated.

Examples of a number $d$ that is a divisor of the order of a group $G$, but there is no quotient group $G/N$ whose order is $d$.

A finite group of order $mn$ with $m,n$ relatively prime, together with subgroups of orders $m, n$.

Example of a non normal subgroup of finite index in an infinite group

Find the minimum number of edges in a graph with $3n+1$ vertices if ...

Showing there is a node in the graph with one and only one edge

Does every $3$-regular bipartite graph have a $4$-cycle?

Classification of groups such that the converse to Lagrange holds

Let G be a non abelian group with a square-free order. Prove that there are two elements $a,b\in G$ such that $ab\neq ba$ and $ord(a)=ord(b)$.

Prove that if $G$ is a finite simple group containing a subgroup $H$ of order $27$, then $|G : H| \ge 9.$

A group such that all its subgroups are finitely generated

Planar graph K3,3

Find the value of an expression related to the roots of a quadratic polynomial

How to find subgroups of a product group containing the diagonal?

How many planar graphs of vertex degrees all ≥4 are there?, extended (by request of G.M.)

An example of a non-diagonalisable matrix in $\mathrm{SL}(n, \mathbb{Z})$ whose eigenvalues don't all have absolute value $1$

If $|G|=n>1$, then $|{\rm Aut}(G)|\le\prod_{i=0}^k(n-2^i)$ for $k=[\log_2(n-1)]$.

Is the intersection of a descending chain of finitely generated groups finitely generated

Number of elements of order coprime to $p$ in a finite group

Prove that : $f(n) \le \frac{1}{4}(n-1)^2+1 $

If $a_n>0$ for all $n$, and $\lim_{n\to\infty}a_na_{n+1}=A$, $\lim_{n\to\infty}a_na_{n+2}=B$, $\lim_{n \to \infty}a_na_{n+3}=C$, dis/prove $A=B=C$

If $0\leq X \leq \text{Id}$ and $0\leq A$, then $XAX \leq A$?

Let $x$ be $(1\dots n)\in S_n$ show that there exists $y\in x^{S_n}$ such that $x^{S_n} = x^{A_n}\sqcup y^{A_n}$.

Confusion in understanding of Steinitz Exchange Lemma by an example

Do these very VERY weak axioms guarantee a group? (Every element has left identity $e_L$ or right identity $e_R$, with same-sided inverse)

Let $G$ be generated by $\begin{pmatrix}1&p\\ 0&1\end{pmatrix},\begin{pmatrix}1&0\\ p&1\end{pmatrix}$ for prime $p$. Is $G$ is solvable? A proof.

Proof that a graph with at most $n+2$ edges is planar

Number of bipartite graphs given two sets