# Tagged Questions

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### what are the other 2 nontrivial elements of the automorphism group of $\Bbb Z/5\Bbb Z$?

It is known that the automorphism group of the units of $\Bbb Z/5\Bbb Z$ is isomorphic to the cyclic group of order $4$, so the automorphism group must also have $4$ elements. The two nontrivial ones ...
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### Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle$$ is cyclic of order 24.
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### Finite groups with a cyclic maximal subgroup.

In the book A Course in the Theory of Groups by Derek J.S. Robinson, Finite $p$-groups with a cyclic maximal subgroup are classified. Now I wish to know whether finite groups with a cyclic maximal ...
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### Inconsistent definition of Sylow p-subgroup

Here is the definition of a Sylow $p$-subgroup from Wikipedia: For a prime number $p$, a Sylow $p$-subgroup (sometimes $p$-Sylow subgroup) of a group $G$ is a maximal $p$-subgroup of $G$, i.e., a ...
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### Cokernel of injective endomorphisms of a finitely generated free abelian group

By $\text{GL}^+(n,\mathbb{Z})$ we mean the set of $n×n$ invertible matrices with positive determinant and entries from $\mathbb{Z}$. For given $A \in \text{GL}^+(n,\mathbb{Z})$ let ...
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### Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
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### For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...
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### Probability of $\alpha\beta\gamma=\gamma\beta\alpha$ for random permutations of a finite set?

Following up on my previous question Probability that two random permutations of an $n$-set commute?, here's a related question for three elements. Q: If $\alpha,\beta,\gamma$ are chosen uniformly ...
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### Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
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### Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
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### Prove that $G$ has a nontrivial normal subgroup

I wanted to ask if I had done this problem correctly. Let $G$ be a group of order $pqr$ (for $p > q > r$ primes). (i) If $G$ fails to have a normal subgroup of order $p$, determine the ...
A Hall system of $G$ is a set $\Sigma$ of Hall subgroups of $G$ satisfying the following two properties: -For each $\pi$ divisor if $|G|$, $\Sigma$ contains excatly one Hall $\pi$-subgroup $G_{\pi}$. ...