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Questions tagged [finite-groups]

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.

The questions that you can ask under this tag are of the same type that the questions related to the tag : homomorphisms of finite groups, representation theory, structure of finite groups...

You may refer to this Wikipedia entry for an introduction to this topic.

11774 questions
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What is the proper background for reading the GLS 'second-generation' proof of the CFSG?

I am aware that this series is incomplete, but it has a large body of existing content, and I am also aware that it is written to be "accessible to non-specialists", but that is obviously quite vague. All that really tells me is that I don't have…
May Emerson
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The relationship between prime divisor of the number of cyclic subgroups of order $2$ with prime divisor of a finite group

Let $G$ be a finite non-abelian group and $k$ be the number of cyclic subgroups of order $2$. Why is every prime divisor of $k$ a prime divisor of the order of $G$?
q-Javadi
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Every conjugate of Sylow permutable is Sylow permutable.

I am working on a proof for the statement: 'If $H$ is a Sylow permutable subgroup of a finite group $G$, then every conjugate of $H$ is Sylow permutable in $G$. ' My current proof is based on the observation that if $K=H^g$ and $P$ is any Sylow…
Khaled
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Generalized permutability and conjugation

Two subgroups $H$ and $K$ of a finite group $G$ are $4$-permutable if and only if $HKHK= \langle H,K\rangle$ (where $\langle H,K\rangle$ is the smallest subgroup of $G$ containing both $H$ and $K$). Is it true that if $H$ $4$-perm $K$, then $H$…
Khaled
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Every $2$-nilpotent group is solvable

Let $G$ be a finite group. If $G$ is $2$-nilpotent then $G$ is solvable. I know a way to prove this result using the Feit-Thompson Theorem: Let $H$ be a normal $2$-complement of $G$. Then $H$ is solvable by Feit-Thompson since $|H|$ is odd and…
nom
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Why don't we consider non-units as quadratic residues?

Is there any specific reason in not including non-units of $\mathbb{Z}_n$ as quadratic residues? As an examples, we say that in $\mathbb{Z}_8$, the set of quadratic residues is just {1} and not {1,4}.
Shiva
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Is the product of two soluble groups always soluble??

I know that there are finite non-soluble groups with soluble subgroups. Is it then possible to produce a non-soluble group from soluble groups?
user862
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Algebra: Order of an element

Suppose that $G$ is finite and that $g\in G$ . If $h\in G$, show that $o(hgh^{-1})=o(g)$. Solution: We have that $(hgh^{-1})^{o(g)}=hgh^{-1}hgh^{-1}...hgh^{-1}=hg^{o(g)}h^{-1}=h1_Gh^{-1}=1_G$, so $hgh^{-1}$ has finite order and $o(hgh^{-1}) \leq…
MathGeek1998
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On the converse of Lagrange's theorem

We know that the order of an element in a finite group must divides the order of the group by Lagrange theorem. Is the converse true ? How do we prove it. I know that the converse is true for nilpotent groups.
user761969
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How I can prove tht $T$ is isomorphic to a finite set of natural numbers?

Let $T$ be a finite abelian group. We can consider $T$ the as group $ℤ/nℤ$ or $ℤ/qℤ×ℤ/mℤ$. My question is: How I can prove tht $T$ is in bijection with a finite set of natural numbers? That is, I want to prove that there is a bijection between $T$…
Safwane
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A question about loss of generality in theorem 2.18 (Zenkov) of Isaacs' Finite Group Theory

Good evening! I was just wondering if somebody could explain how we are able to make an assumption (that I will mention below) in the following theorem (2.18 in Isaacs' Finite Group theory; it's attributed to Zenkov): Let A and B be abelian…
TenaxPropositi
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Q question about proving isomorphism of abelian groups

Suppose that $\mathbb{Z}_n^{+}$ denotes the cyclic group of order $n$. Question a: Consider the group $$ G=\mathbb{Z}_{n_1}^{+}\times \mathbb{Z}_{n_2}^{+}\times \ldots \mathbb{Z}_{n_k}^{+} $$ where $n_1,\ldots,n_k$ are pairwise relatively prime.…
boaz
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Elements of Order 6 in a simple group of order 168

A similar question has been asked before but I believe that this question is slightly different. So far,the analyses of simple groups of order 168 have no elements of order 6, need to rely on computing the number of Sylow 3-subgroups, $n_3$. (Such…
daruma
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On the center of a finite group $G$ with a normal Sylow subgroup

Suppose nonsolvable finite group $G$ has a normal Sylow $p$-subgroup. I would like to know whether center of the group $G$ is nontrivial?
Simon
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How to find number of homomorphisms between z8 and s4?

How to find number of homomorphisms between z8 and s4? I check by taking image of generator of z8 Say f:z8---->s4 O(f(1))/8 O(f(1))=1,2,4 S4 has 15 elements of order 2 and 6 elements of order 4 15+6+1(trivial)=22 Is this correct?
Radhika Jain
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