# 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.

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### Centraliser of a finite group

Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$. We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
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### A modification of a cyclic group that seems to break being a group. What is it?

Background: I came up with this trying to answer an actually silly question of "when can $1+1=3$ be true" ? Consider a set $\mathcal S = \{0,\cdots, N-1\}$ coupled with an operation "+&...
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### Why is a Sylow 5-subgroup abelian?

For weeks I tried to solve the following question on Brilliant: Fill in the blank: "Every group of order ___ is abelian." And these are the possible answers I get: 15, 16, 20, 21, 27. Using ...
1 vote
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### "Simple" group of order $1004913$ problem, fixed point part

Let $G$ be a group of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$. We suppose that $G$ is simple. We want to obtain a contradiction. This is the Exercise 29 in Chapter 6.2 of Dummit-Foote. As ...
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### About Frobenius Groups of order 1029

In the list of groups in GAP of order $1029=7^3\cdot 3$, there are two, with structure description $U_3(\mathbb{F}_7)\rtimes C_3$. (Among $19$ groups $G, G, \ldots, G$ of order $1029$, the ...
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### Constructing central extensions or Schur cover of U4(3) in GAP

Part of my group theory project involves we're looking at the group $U_4(3)$, which has abnormally large Schur Multiplier (36) and large automorphism group ($D_{12}$). I need to work with the central ...
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### Character tables of isomorphic groups

Let us consider two groups $G$ and $G'$ which are isomorphic to each other. Since isomorphic groups can be considered as same upto isomorphism, is the character table same for both groups?
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### Find an element of order $p+1$ in $GL_2(\mathbb{Z}/p\mathbb{Z})$ [duplicate]

The question says it all. I'm looking for an element of order $p+1$ in $GL_2(\mathbb{Z}/p\mathbb{Z})$. I have already computed the order of the group (so I know it's possible that one exists). I have ...
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### How to show that a subgroup of a product of finite cyclic group has itself this property?

Let $G$ be a subgroup of a finite product of finite cyclic groups. Is it easy to prove that $G$ is itself a finite product of cyclic groups without appealing to the structure theorem of finite abelian ...
1 vote
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1 vote
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### Modules over the ring $\mathbb{F}_p[C_p]\cong \mathbb{F}_p[X]/(x^p-1)$

I would like to understand the category of modules over the group algebra $\mathbb{F}_p[C_p]\cong\mathbb{F}_p[X]/(x^p-1)$. I am interested in computing the group cohomology of $C_p$ with coefficient ...
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### In a group of order 120 with a normal subgroup of order 5, every subgroup of order 15 contains that normal subgroup. Simplest proof?

Let $G$ be a group of order $120$ with a normal subgroup $N$ of order 5. Let $H$ be any subgroup of $G$ of order $15$. Prove $N$ is a subgroup of $H$. I have a proof (see below), but I am wondering: ...
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### Doubt regarding the generality of an equivalent definition for Abelian Group [duplicate]

I was given a problem as follows: Let $G$ be a finite group of odd order. If $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b\in G$ then $G$ is abelian. I was able to show that this is ...
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### Problem 1D.2 from Martin Isaacs's Finite Group Theory

I try to solve the following problem: Fix a prime $p$, and suppose that a subgroup $H \subset G$ (of a finite group $G$) has the property that $C_G(x) \subset H$ for every element $x \in H$ having ...
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### Find a subgroup $K$ to complete the pullback diagram $G/g_1Hg_1^{-1}\leftarrow G/H\to G/g_2Hg_2^{-1}$.

EDIT: I have realised I made a mistake when decompsoing the morphisms of $\mathscr B_G$. Nevertheless, the question seems to be interesting on its own, so i will leave it. I would also like to cite ...
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### Frattini subgroup of a quotient

Let $G$ be a finite group. The Frattini subgroup $\Phi(G)$ is the intersection all proper maximal subgroups. If $K \lhd G$ is a normal subgroup, then it is easy to see that $\Phi(G) K/K \leq \Phi(G/K)$...
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1 vote
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### Lyndon-Hochschild-Serre for the product

General case I'm interested in: let $G = H_1 \times H_2$ where both $H_i$ are abelian groups. Let $M \in \operatorname{Mod}_{k[G]}$, then one has the spectral sequence $H^i(H_1, H^j(H_2, M))$ ...
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### How many non-isomorphic groups of order $5832 = 2^3 \cdot 3^6$ are there?

I'm afraid I can't provide much motivation other than personal interest. I have found David Burrell's very recent Ph.D. thesis, which identified a transcription error that resulted in an incorrect ...
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### Group algebra is a free module of finite rank over any group sub-algebra

Let $G$ be a finite group and let $H$ be a subgroup. Let $\mathbb CG$ and $\mathbb CH$ be the corresponding group algebras. Clearly, the injective group homomorphism $H\to G$ defines a $\mathbb CH$-...
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### Finite group generated by two finite subgroups

Question Let $G=\langle H,K\rangle$ where $H,K$ are two finite subgroups of $G$. If $H$ is subnormal in $G$, then show that $G$ is finite. Attempt I know that $D_{\infty}=\mathbb{Z}_2\ast\mathbb{Z}_2$ ...