The study of symmetry: groups, subgroups, homomorphisms, actions

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Group of infinite sequences of elements in a given finite group

Could you tell me why each element of a group of infinite sequences of elements in a given finite group has finite order?
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$\mathbb Z_p^*$ is a group.

I'm trying to prove that $\mathbb Z_p^*$ ($p$ prime) is a group using the Fermat's little theorem to show that every element is invertible. Thus using the Fermat's little theorem, for each $a\in ...
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$G$ is a nonabelian finite group, then $|Z(G)|\leq \frac{1}{4}|G|$

If $G$ is a nonabelian finite group, then I have to show that $|Z(G)|\leq\frac{1}{4}|G|$, where $|Z(G)|$ denotes the center of group. I have got this question through random search on net. I am ...
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Either all elements of a subgroup of $\mathbb{Z}_n$ are even or exactly half of them

Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb{Z}_n$. Prove that either every member of $H$ is even or exactly half of the members of $H$ are even. Well, I know that if $0 ...
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Which elements generate $\mathbb Z \oplus \mathbb Z$

It's a simple question, but I have a little problem with an aspect of the proof. I'm trying to prove that this set $\{(1,0),(0,1)\}$ generates $\mathbb Z \oplus \mathbb Z$. Following the definition ...
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There exist Sylow subgroups $P$ and $Q$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$.

From D&F's sylow theory section: Show that if $n_p\not\equiv 1 \mod p^2$ then there are distinct Sylow $p$-subgroups $P$ and $Q$ of $G$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$. Are ...
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How to determine centralizers of $A_5$?

Prove that the order of a conjugacy class of $A_5$ is $1,12,15$ or $20$. There are 4 different cycle type in $A_5$. There are 20 permutations of cycle type $(123)$, 24 permutations of cycle ...
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Can a group of order $55$ have exactly $20$ elements of order $11$?

Can a group of order $55$ have exactly $20$ elements of order $11$? Give a reason for your answer by sylow theorem the answer is easy but without using sylow how can I solve this.can anyone help me ...
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Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
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Let $G$ a finite group. And let $\varphi : G→S(G):g↦λ_g$. If $|G|=nm$ and $|x|=n$, then $\varphi(g)$ is the product of $m$ disjoint $n$ cycles.

Let $G$ a finite group. And let $\varphi : G→S(G):g↦λ_g$. If $|G|=nm$ and $|g|=n$, then $\varphi(g)$ is the product of $m$ disjoint $n$ cycles. When writing this question, I think I've proven ...
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List of all elements of the Weyl group of type $C_3$.

What is the list of all elements of the Weyl group of type $C_3$ in terms of simple refletions $s_1, s_2, s_3$? There are 48 elements in the group. Thank you very much.
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on a group with perfect automorphism group

A group $G$ is called perfect if $G=G'$. Does there exist a group $G$ such that $Aut(G)$, the automorphism group of $G$, is perfect?
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1answer
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Studying the action of $GL(V)$ on the vector space $V$

The statement I am trying to prove is the following. Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let $\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural ...
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Why is $| \rm{Aut}(\mathbb{Z}_n) | = \phi(n)$?

If $G$ is a finite cyclic group of order $n$, prove that $| \rm{Aut}(G) | = \phi(n)$, where $\phi(n)$ is the Euler's totient function. Can someone please help me with this?
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(Hypothetical) simple group of order 1004913

In 6.2.29 of Dummit and Foote, the exercise is to prove a group of order $3^3*7*13*409$ is not simple by using the permutation representation of degree 819. After having calculated the uniquely ...

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