# Tagged Questions

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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### Let $H/N$ is nilpotent quotient subgroup. Then is $N$ characteristic in $H$? If no, then what condition need?

Let $G$ is finite solvable group and $H$ is normal subgroup. Let $H/N$ is nilpotent quotient subgroup. Then is $N$ characteristic in $H$? If no, then what condition need?
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### Action of ${\rm Aut}(G)$ on $G$

Let $G$ be a finite group and consider the natural action of ${\rm Aut}(G)$ on $G$ and let there are two orbits under this action. How could we show that $G$ is an (elementary) abelian group? Is ...
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### Problems with P. Hall theorem proof (The problem involves the use of Frattini's argument)

Theorem Let G be a solvable group of order $ab$, where $(a,b)=1$. Then $G$ contains at least one subgroup of order $a$, and any two such are conjugate. Details The proof the book presents involves ...
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### How many subgroups are there in an elementary-$p$ group

$C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally. But how many subgroups in $C_p\times C_p \times C_p\times C_p$? (...
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### p-group of class 2 with cyclic commutator subgroup

I was reading a paper "ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS" you can get it from here. In Introduction page 2 He said "It is worth mentioning here that we need ...
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### G's order is a multiply of coprime numbers, need to prove about its subgroups.

dont want you to answer me directly, only a direction of thinking. I have abelian $G$ of finite order $np : p>n, p>1,$ and p is prime. $A,B\le G$ are sub-groups of G of order $p$ both. I need ...
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### $G = MC$ for some cyclic subgroup $C$. If $\vert G : M \vert = 4$ then why $G \cong S_{4}$?

Let $M$ is a maximal subgroup of finite group $G$, that $G = MC$ for some cyclic subgroup $C$. If $\vert G : M \vert = 4$ and $M_{G} = 1$ then why $G\cong S_{4}$?
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### $G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then $G/G' \cong \widehat G$?

Let $G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then is it true that $G/G' \cong \widehat G$ ? I am completely stuck , please help . ...
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### Pretty easy equations of elements in a group

Problem $G$ is a group generated by $a,b\in G$ such that $a^5=e$, $aba^{-1}=b^2$ and $b\ne e$. I want to find the order of $b$. Attempt I tried to multiply the second equation from right by $a^{4}$:...
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### Geometric or at least application view of a group with 3 elements?

My teacher asked us to find applications in real life, or ways that a group with 3 elements migth show up in real problems, and the one I gave was about an watch for a planet where the entire day ...
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### Understanding a proof of a lemma to Jordan-Hölder Theorem.

I have difficulty understanding the following lemma. First, how do we know that $M\cap N={1}$, after replacing $M$ and $N$ as in the proof? Next, in the second part of the proof where it says we ...
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### Center of a maximal subgroup

Is it possible to find a non-abelian group (preferably a finite $p$-group) with the following property? If M is a maximal subgroup of a group (or finite $p$-group) $G$, then $Z(M)\not \le Z(G)$.
### for prime $q\ge p$, integer $m\ge 0$, any group with order $p^2 q^m$ is not simple
I tried the two ways, but both are all failed. First, counting the order of union of all Sylow $p$, $q$-subgroup. Second, group action from orginal group to set of cosets by Sylow subgroup. Is there ...
This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Denote the finite matrix group $\{\pm I_2\}\subset GL(2,k)$ by $C_2$. It is known that the invariant polynomials under $C_2$ ...