# 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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### No simple group of order 756 : Burnside's proof

I'm interested in a proof of the non-simplicity of groups of order 756. W.R. Scott, Group Theory, p. 392, exerc. 13.4.9, gives it as an easy exercise, but depending on rather advanced results. I have ...
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### "Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?

I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
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### Use of correspondence theorem for groups to prove that $o(N) = 2$

Let $G$ be a group and $H \triangleleft G$ simple such that $[G : H] = 2$. I have to prove that if $N \neq \{1\}$, $N \triangleleft G$ and $N \cap H = \{1\}$ then $o(N) = 2$. I know by third ...
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### Proof that a group of order $180$ is not simple without Burnside p-complement theorem

A proof that a group of order $180$ is not simple is given here. However, the proof uses Burnside $p$-complement theorem. If you know a proof without Burnside $p$-complement theorem, please let me ...
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### On the number of invariant Sylow subgroups under coprime action - Antonio Beltrán and Changguo Shao article

This is an article that Antonio Beltrán and Changguo Shao wrote. Lemma 2.5. states: [All groups are supposed to be finite (this is mentioned before)] Lemma 2.5. Let $A$ be a group acting coprimely on ...
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### Prove the index of a proper subgroup of a simple group of order 17971200 is at least 14.

I didn't find a solution for this problem or other usual approaches that could directly work. So, here is my attempt. I am self-studying and reviewing group theory recently, and would like to know if ...
1 vote
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### Isomorphic two abelian subgroups which lies in the some finite union of conjugacy classes of simple group [closed]

Let $G$ be a simple group and $H_1$ and $H_2$ be abelian subgroups of $G$ such that $H_1\cong H_2$ and $H_1,H_2\subseteq Cl_G(e_G)\cup Cl_G(x)$ for some $x\in G$, where $Cl_G(\cdot)$ denotes the ...
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### "$S$ is the unique irreducible $\operatorname{End}(S)$-module" [duplicate]

Let $S$ be a finite-dimensional vector space. Then $S$ is an irreducible $\operatorname{End}(S)$-module. Furthermore I was told that $S$ is the only irreducible $\operatorname{End}(S)$-module. I guess ...
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### An example of non simple group which is also a Lie group such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup

I want to know if there exist a non simple group (as abstractl group) $G$ such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup. I have tried some obvious examples like ...
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### Let $K\lhd G$ be s.t. both $K$ and $G/K$ are simple. Show that either $K$ is the only proper normal subgroup of $G$, or $G \cong K \times (G / K)$.

Sorry about the title, I couldn't fit the whole exercise (Exercise 8.1.6, Nicholson Introduction to Abstract Algebra 4th edition): Let $K \triangleleft G$ be such that both $K$ and $G/K$ are simple. ...
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### No simple group of order $p^nq^m$, with barely invoking Sylow theorems

It is a well known fact that for two distinct primes $p$ and $q$, and natural numbers $m, n \geq 1$, there can be no simple group of order $p^nq^m$. Most proofs I have seen of this statement either ...
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### Is there an explicit maximal simple group?

It is not hard to prove the following lemma: LEMMA: Let $(G_i)_{i \in I}$ be a chain of simple groups. Then $G = \bigcup_{i \in I} G_i$ is a simple group. Let $N$ be a normal subgroup of $G$ and ...
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### Show that $SL(2,5)$ has no subgroup isomorphic to $A_5$

I'm trying to show that there's no subgroup of $SL(2,5)$ isomorphic to $A_5$. I've already shown that $A_5$ is simple, and my strategy is to show that $SL(2,5)$ contains no simple subgroups of ...
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### Finite non abelian simple groups whose order is not divisible by 8

Is $A_5$ the only finite non abelian simple group whose order is not divisible by 8? If not, what is the complete set of finite non abelian simple groups whose order is not divisible by 8?
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### Simplicity of the $A_5$ [duplicate]

In the proof given in Abstract Algebra by Dummit and Foote, page 128, it states that $(1~2)(3~4)$ commutes with $(1~3)(2~4)$ but does not commute with any element of odd order in $A_5$. So it follows ...
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### Let $G$ be a finite simple non abelian group and $\{1\}\neq H\le G$ be such that $\vert C_{G}(H)\vert=\vert G:H \vert$. Then $H=G$.

Let $G$ be a finite simple non abelian group and $\{1\}\neq H\leq G$ be such that $\vert C_{G}(H)\vert=\vert G:H \vert$. Show that $H=G$. I can find only $[G,G]=G$ and $Z(G)=\{1\}$ and if $H$ is a ...
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### Is every finite simple group a quotient of a braid group?

Question: Is every finite simple group a quotient of a braid group? Context: The braid group on two strands $B_2$ is isomorphic to $\mathbb{Z}$ and so the infinite family of abelian finite simple ...
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### There are no simple groups of order $480$

Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is 480; this self-answered question aims to fill that gap. (...
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### There are no simple groups of order $336$

Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $336$; this self-answered question aims to fill that gap....
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### There are no simple groups of order $264$

Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $264$; this self-answered question aims to fill that gap....
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