# Questions tagged [simple-groups]

Use with the (group-theory) tag. A group is simple if it has no proper, non-trivial normal subgroups. Equivalently (for finitely generated groups), its only homomorphic images are itself and the trivial group. The classification of finite simple groups is one of the great results of modern mathematics.

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### $[G:H]=4$ where $H\neq 1$ means that $G$ is not simple.

The question is as follows: Suppose $G$ is a finite group with a nontrivial subgroup of index $4$. Prove that $G$ is not simple. I am on the hunt for a non-trivial normal subgroup. This proof ...
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### Diagonal in the power of a group

Let $L$ be a simple group and $G=L^t$. The diagonal $D=\lbrace(x,x,\ldots,x), x\in L\rbrace$ is a subgroup of $L^t$. I can prove that if $D$ is maximal then $t$ is a prime. If $t=mn$ then the subgroup ...
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### Are commutator subgroups simple?

I am interested in the following setting: $$0\rightarrow [G,G]\rightarrow G \rightarrow \mathbb{Z}^r\rightarrow 0,$$ in particular in the case $r=1$. Is $[G,G]$ simple in this case? I am aware of ...
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### If $G$ is simple, then $Z({\rm Aut}(G)) = \{1\} \iff G$ is non-abelian [duplicate]

NOTE: I am aware of the link at For a Simple Group G, Z(Aut(G)) Is Trivial if and only if G is Non-Abelian. I am looking for proof verification of $G$ non-abelian $\implies Z(\mathrm{Aut}(G)) = \{1\}$....
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 ...
1 vote
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### A reference request for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$.

Note: This is a reference-request question and thus does not need the usual type of context. The Question: What is a reference for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$? Background: ...
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### Has every infinite simple group a faithful irreducible representation?

Has every infinite simple group a faithful irreducible representation? This question solves the finite case. However, the proof requires a non-trivial linear representation of a finite group. I want ...
1 vote
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### Is there a way to computationally verify that the sporadic groups are simple?

I'm trying to understand the "easy" direction of the CFSG: namely, the proofs that the 18 infinite families and 26/27 sporadic groups are indeed simple. I'm working through Simple Groups of ...
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### The only proper normal subgroups of a nonabelian quasisimple group are subgroups of its centre.

This is part of a bunch of exercises set by my academic supervisor. As such, I'm not sure whether all the hypotheses are needed for the conclusion. Please note that hints are preferred. The Question: ...
1 vote
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### Does every perfect finite group have a fixed point free representation?

Fixed point free representations of finite group are important for the spherical space form problem and also show up in other contexts for example Perfect semi direct products A representation $\pi$ ...
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### To what extent can we determine the simplicity or non-simplicity of groups based on their prime decompositions?

This question may have more of a vague, less objective answer than usual for this site, so I apologise if it difficult to answer definitively. Below, $p$, $q$ and $r$ are distinct primes. A group of ...
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### Every proper subgroup of a simple group containing an order 45 element has an index of at least 14

Every proper subgroup of a simple group containing an order $45$ element has an index of at least $14$. So far I've supposed that $G$ is a simple group with an element of order $45$. Then $45$ ...
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### Automorphisms of $SL(n,q)$ coming from $GL(n,q)$

For which values of $n \geq 2$ and $q$ a prime power are all automorphisms of $SL(n,q)$ induced by conjugation by elements of $GL(n,q)$? Old version of the question: $\Sigma L(n,q)$ is an ...
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### Natural group extension constructed from Schur cover and its outer automorphism group

Let $S$ be a finite (non-abelian) simple group. Then there always exists a natural extension of $S$ by the outer automorphism group $Out(S)$ with elements of $Out(S)$ acting as outer ...
1 vote
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### Nontrivial extension of cyclic group by simple group

Let $G$ be a (non-abelian) finite simple group. An extension $G\cdot m$ is nontrivial if it is not isomorphic to the direct product $G \times m$. Suppose that there exists a nontrivial extension ...
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### No simple group of order 1040

Burnside (Proceedings of the London Mathematical Society, vol. 26; Collected Papers, vol. 1, p. 601) gave the following proof of the non-simplicity of groups od order 1040 : "If simple, the group ...
1 vote
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### Show that a finite simple group $G$ of even order must have order divisible by $4$.

It is requested to proof exercise $3.30$, $(ii)$ from J. Rotman, An introduction to the theory of groups, which states that a finite simple group $G$ of even order greater than $4$ must have order ...
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### Equivalent of simple groups in topology

I read about the classification theorem of finite simple groups, and I was wondering if there is a topological meaning to spaces that have a simple fundamental group? In addition, is there something ...
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### On Subgroups of Same order of Simple group

We know that, If there exists none other than only subgroup $H$ of order $m$, then $H$ is normal in $G$. When It comes to simple group, There should exist at the least $2$ subgroups of same order. To ... 1 vote
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### Image of minimal degree representation of quasisimple group unique up to conjugacy.

Let $G$ be a quasisimple finite group. Let $d_{min}$ be the minimum dimension of a nontrivial irrep of $G$. Must it be the case that the image of all (nontrivial) dimension $d_{min}$ irreps of ...
1 vote
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### What is the relation of word "simple" with math-word "simple group" of group theory?

Is there any relation with the meaning of word "simple" with what "there are groups $G$ in which the only normal subgroups are the trivial ones: $1$ and $G$. Such groups are called ...
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### Is every finite non-abelian simple group generated by involutions?

Let $G$ be a finite non-abelian simple group. Is it true that the set of involutions $$\{ g: g\in G, g^2=1 \}$$ generates $G$? For example consider the group $G=A_5$ of order $60$. The ...
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### Why is studying centralizers the/a key to classifying finite groups?

In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says One problem, as least with the current methods of ...
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### Why do we trust the Classification of Finite Simple Groups?

It seems to me there are a two main reasons to believe a theorem/conjecture to be true: Because it has a correct proof (e.g. the Feit-Thompson Theorem, Dirichlet's Theorem) Because there is an ...
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### Let $G$ be a simple group. Show that any homomorphism from $G$ to $G'$ (arbitrary $G'$) must be either injective or the trivial homomorphism. [duplicate]

I've spent a lot of time thinking about this question but I can't seem to come up with anything of substance. I tried to use some basic lemma for group homomorphisms, but nothing seemed to work. My ...
1 vote
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The following is an exercise, to prove simplicity of a group $G$, which satisfies conditions - i) $G$ is transitive subgroup of $S_p$ (symmetric group, $p$ is prime) ii) $|G|=p\cdot m\cdot k$, iii) $m\... 0 votes 0 answers 50 views ### Statement explanation of a paper about finite simple groups I was trying to understand the statement of the main theorem of this paper https://arxiv.org/pdf/1510.03665.pdf The statement says that if$S$is a finite non-abelian simple group and$r$is an odd ... 0 votes 1 answer 48 views ### Are complex reflection groups never perfect? This is a follow-up to Conceptual reason why Coxeter groups are never simple A complex reflection group is a finite subgroup of$ U_n $that is generated by pseudo reflections. A pseudo reflection is ... 1 vote 1 answer 72 views ### Group mod center is perfect structure result Suppose that $$1 \to Z(G) \to G \to P \to 1$$ is a short exact sequence of groups where$ P $is perfect and$ Z(G) $is the center of$ G $. Must it be the case that$ G $is the direct product of ... 2 votes 1 answer 151 views ### A question about the classification of compact Lie groups I was able to prove that every compact Lie group is isomorphic to $$(T^n\times G_1\times \cdots \times G_m)/\Gamma$$ where the$G_i$are compact, simply connected, simple Lie groups and$\Gamma$is a ... 2 votes 1 answer 169 views ### Prove that$A_5$is simple. Prove that$A_5$is simple. The solution presented is as follows: Let$N$be a non trivial normal subgroup of$A_5.$If$(12)(34)\in N,$then $$(25)(34)(12)(34)(34)(25) = (15)(34)$$ and$$(12)(34)(15)... 0 votes 1 answer 113 views ### For$n\ge 5,\forall e\neq\rho\in A_n,\exists\sigma=\alpha\rho\alpha^{-1}$for some$α\in A_n$s.t.$ρ(i)=σ(i)$for some$i\in\{1,2,...,n\}$, but$ρ≠σ$For$n\ge 5$, for all$e\neq \rho\in A_n, \exists \sigma=\alpha\rho\alpha^{-1}$for some$\alpha\in A_n$such that$\rho(i)=\sigma(i)$for some$i\in\{1,2,...,n\}$, but$\rho\neq\sigma$. The solution ... 2 votes 0 answers 59 views ### Is every finite group a subgroup of this type of generalization of almost simple groups? Recall, a group$G$is an almost simple group if there is an non-abelian simple group$S$such that$S \leq G \leq \operatorname{Aut}(S)$We will define a$k$-almost simple as follows. When$0$--... 2 votes 0 answers 55 views ### random generation of lie algebras It is well known that a nonabelian finite simple group, say$\mathrm{PSL}_n(\mathbf{F}_p)$, can be generated by two elements. In fact, the probability that two elements generate it tends to$1$as the ... 1 vote 1 answer 107 views ### Why can't a group of order 132 contain 3 Sylow$2$-subgroups and be simple? In the question titled Prove that if |G|=132 then G cannot be simple, it is shown a group of order$132$cannot be simple. My summary of the proof: assume group is simple deduce the number of Sylow$...
This question arose from my attempt at understanding the answer in this post - the comments below it, to be precise. Everything revolves around the following problem: Let $S$ be a simple, nonabelian ...
My goal is to prove the following: Let $G$ be a finite group and let $S \leq G$ be a simple subgroup. Suppose that $SH = HS$ for all subnormal subgroups $H$ of $G$. Show that $S$ is contained in the ...