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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Existence of a finite simple group that satisfies particular properties.

Let $G$ be a finite simple group and $\tau_G = \{ o(x) : x \in G\}$. Does there exist $d_1, d_2 \in \tau_G$ that satisfy the following: $d_1 < d_2$ and $d_1$ does not divide $d_2;$ for $x, y \in ...
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Restrictions on the order of Finite Simple Groups

I've just started learning about simple groups and I'm curious about restrictions on the order of finite simple groups. For example, I know that the only abelian finite simple groups are cyclic with ...
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GAP Program Efficiency for Number of Orbits

Let $S$ be a nonabelian finite simple group and $p$ a prime divisor of $|S|$. I'm interested in finding the number of $\textrm{Aut}(S)$-orbits acting on the set $\textrm{Cl}_{p'}(S)$, the set of all $...
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A finite nonabelian group is not simple if any two of its elements who are conjugate to each other commute.

Problem: Let $G$ be a finite nonabelian group. Assume any two elements $x,y \in G$ conjugate to each other also commute, show that $G$ is not simple. I write down a proof of this problem based on its ...
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Is this inverse image nontrivial?

I would like to prove that if a non-comutative group $G$ has a subgroup $H$ whose index is equal to $3$ or $4$, then $G$ is not simple. In order to do, I took a group action $$\phi : G \times G/H \ni (...
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Schur multiplier of almost simple group

Let $S$ be a finite simple group. We call a finite group $G$ an almost simple group with respect to $S$ if $S\leq G\leq \mathrm{Aut}(S)$. For a finite simple group, $\mathrm{M}(S)$, the Schur ...
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On nonsplit noncentral extension of finite simple groups

Let $G$ be a finite group and $N$ be its minimal normal subgroup such that $G/N$ is a finite simple group. It is well know that if $G=G'$ and $N=Z(G)$, then $N=\Phi(G)$. My Question is: Is it true ...
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What is the intuition behind simple Lie groups?

What is the intuition behind simple Lie groups? Background: Simple groups and their final-dimensional representations are one of the huge improtance topics at my university course. My questions: Why ...
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Is the Socle of an almost simple group a simple group?

Let $G$ be a finite primitive group of degree $n$, and let $H$ be the socle of $G$. Then if $H$ is isomorphic to a direct power $T^m$ of a nonabelian simple group $T$ then the following holds when $m=...
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Let $G$ be a non abelian simple group. Show that ${\rm Aut}(G^n)$ is isomorphic to ${\rm Aut}(G) \wr{\rm Sym}(n).$

I got this question but don't know how to answer it. Let $G$ be a non abelian simple group. Show that ${\rm Aut}(G^n)$ is isomorphic to ${\rm Aut}(G) \wr{\rm Sym}(n)$. I already know that ${\rm Aut}(G)...
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The compactness theorem for simple groups

I would like to use the compactness theorem to prove the following: Claim. If $H$ is a subgroup of countable index in a simple group $G$, then there are finitely many conjugates of $H$ with trivial ...
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Proof there are no perfect groups of order 3024

How can I prove that there are no perfect groups of order $3024$? My attempt is the following: Each non-trivial finite perfect group admits a non-abelian simple quotient. This holds because if the ...
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Minimal normal subgroups of Product of simple groups

Known Result: Let $$G= S_1 \times S_2 \times\dots\times S_n,$$ where each $S_i$ are non-abelian simple groups. Then $S_i$'s are the minimal normal subgroup of $G$. (Even $S_i$'s are the only minimal ...
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No group of order 160 is simple [duplicate]

I'm trying to prove that no group of order 160 is simple. The following is my approach. Let $G$ be a group of order $160$. Note $160 = 2^55$. I can easily get that $n_2 = 1$ or $5$ and $n_5 = 1$ or $...
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If $S$ is a simple subnormal subgroup of $G$. Prove that if $S$ is nonabelian then $S^G$ is a direct product of simple groups isomorphic to $S$.

Here is the question and my solution. I understood the answer discussed here. My question and the solution is slightly different. Which does not use that $T$ is non abelian. Proof : CLAIM-1: $T^G$ is ...
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Group of order $1320$ is not simple

Group of order $1320 = 2^3\cdot 3\cdot 5\cdot 11$ is not simple. Proof. Suppose there is a simple group $G$ of order $1320$. Then the number of Sylow $11$ subgroup $n_{11} = 12$. Let $G$ act on ${\rm ...
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Showing Factor Groups are Simple; Subgroups of Solvable Groups are Solvable

I'm aware this post is related to many other posts on proving that subgroups of solvable groups are solvable. However, there's a certain claim that is necessary to show based on the definition of ...
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If $n\geq 5$ prove that $A_n$ is the only nontrivial normal subgroup in $S_n$. But please don't use the fact $A_n$ is simple if $n\geq 6$.

I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein. The following problem is Problem 17 on p.81 in this book. Problem 17: If $n\geq 5$ prove that $A_n$ is the only nontrivial ...
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Does every quasisimple finite group have a faithful complex irrep?

Every simple finite group has a faithful complex irrep. Indeed any nontrivial irrep of a simple finite group is faithful. That leads me to ask: Does every quasisimple group have a faithful complex ...
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There is no simple group of order $36$.

I tried to do this as an exercise and wanted to ask if my proof is correct or if it is missing something. Thank you so much. Let $G$ be a group such that $\lvert G \rvert = 36 = 2^2 \cdot 3^2.$ Show ...
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Finding a generating set of the simple group of order 168

I'm looking at the symmetries of some geometric object which I think should be the simple group $G$ of order 168. I have in my possession symmetries $\alpha$, $\beta$, and $\gamma$ of orders $2$, $3$, ...
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Is $GL_n(\mathbb{F}_2)$ generated by a Jordan cell of size $n$ and its transpose?

Let $n\geq 3$. Consider the group $GL_n(\mathbb{F}_2)$, and let $J_n=\pmatrix{1 & 1 & & \cr & 1 & 1 & & \cr & & 1 & 1 \cr & & & \ddots&\ddots}$ ...
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Which finite groups can be proved simple via this theorem?

The paper An Elementary Proof of the Simplicity of the Mathieu Groups $M_{11}$ and $M_{23}$ by Robin J. Chapman proves the following theorem: Let $G$ be a transitive subgroup of $S_p$, and suppose $|G|...
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If a chief factor of a group $G$ is not simple, then $G$ is not supersolvable. Is this true? If not, what is needed to make it true?

The Details: A normal subgroup $N$ of a group $G$, written $N\unlhd G$, is a subgroup $N\le G$ such that for all $g\in G$, we have $$gN=Ng.$$ A chief series of a group $G$ is a finite set of normal ...
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Is there a reason the prime factors of $|M_{24}|$ are all one less than the factors of $24$?

Wikipedia says of the Mathieu group $M_{24}$, a $5$-transitive permutation group acting on $24$ points, $$ |M_{24}|= 2^{10}\cdot3^3\cdot5\cdot7\cdot11\cdot23. $$ The prime factors $2,3,5,7,11,23$ are ...
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Growth of finite simple groups

Background. There is an important group-theoretic notion of growth rate, defined for finitely-generated groups $G$ equipped fixed finite generating sets $S$. The growth rate is (the equivalence ...
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If $|G| = n < 60$, and $n$ is composite, then $G$ is not a simple group. Why? [duplicate]

If $|G| = n < 60$, and $n$ is composite, then $G$ is not a simple group. I am not totally sure how to solve this. So far, I have tried thinking of every possible theorem I can think of. A simple ...
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Proving that the symplectic group $Sp_{2n}(K)$ is simple if $|K|=2$ or $3$

I tried to prove the simplicity of $G=Sp_{2n}(K)$ ($n\geqslant 2$), where $K$ is any field. If $|K|\geq 4$, then this is fairly easy to prove. To start, we use the following result that is also used ...
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2 answers
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Assume $H,G$ are simple groups. Can we prove that either $H\unlhd HG$ or $H\unlhd GH$?

Assume $H,G$ are simple groups. Can we prove that $H$ is normal in $HG$ or in $GH$? Context: I want to use this argument in order to prove the following statement: If $H_i$ and $G_j$ are simple such ...
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2 votes
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Proving that the Spinor-kernel of SO3(K) is simple

EDIT: This whole proof fails, as not everything in $SO_3(K)$ can be written as $\sigma_1\sigma_2$. Instead, we have $SO_3(K)\cong SL_2(K)$, which is not simple if $|K|=3$. I tried to prove something, ...
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1 vote
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How is $C_4$ constructed from the simple factor groups of its decomposition series

I often hear that a every finite group is build up from simple groups (specifically the factors in its decomposition series, which are unique by Jordan-Hölder). The cyclic group of order $4$ has a ...
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No simple group of order $2025$

There is no simple group of order $2025$. If $G$ is simple then by the Sylow theorem, there must be $81$ Sylow $5$-subgroups and $25$ Sylow $3$-subgroups. Also, I can see that if Sylow $5$ subgroups ...
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If $G$ is a simple group of order 168, $n_3=28$, then a Sylow 3-subgroup $T$ of $N_G (P_3)$ acts transitively on the set of Sylow 7-subgroups of $G $

I am looking at Abstract Algebra, 3rd ed., by Dummit and Foote, page 208. In classfying groups of order $168$, we first assume that there is a simple group of order $168$ and prove that (1) $G $ has ...
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2-generated finite non-Abelian simple groups and the existence of Hamiltonian cycles in their Cayley graph

Given that $G = \langle a, b\rangle$ and that $a$ is an involution, when is it the case that there exists $c, d$ such that $G = \langle c, d\rangle$ and $cd$ is an involution? At present, I am ...
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Number of conjugacy classes of elements of order $7$ in a group of order 168

Let $G$ be a simple group of order $168$ (Here, we don't assume we know there is a unique such group). Compute the number of conjugacy classes of elements of order $7$ in $G$ [Hint: Consider Sylow $3$-...
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Let $G$ be a simple group, $|G|=n, p$ a prime such that $p|n$. If $G$ has more than $n/(p^2)$ conjugacy classes then the $p$-Sylow sub are abelian

Let $G$ be a simple group, $|G|=n$, $p$ a prime such that $p\mid n$. Prove that if $G$ has more than $n/(p^2)$ conjugacy classes then the $p$-Sylow subgroups are abelian. I think I have to use Sylow's ...
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Show that $A$ is a simple group. [duplicate]

The question is as follows: Let $S_{\mathbb{N}}$ be the group of all permutation of $\mathbb{N}$. Let $F$ be the subset of $S_{\mathbb{N}}$ consisting of all permutations that fix all but a finite ...
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A detail in the proof that a group of order 168 is simple

I am doing an excercise from Gallian's Contemporary Abstract Algebra that wants me to prove that the group $PSL(2,Z_7)$ is simple. I have reached the point $n_2=7$, $n_3=7$ or $28$ and $n_7=8$. I ...
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Intersection of normal subgroups and their indices exercise

Suppose $G$ is a finite group, $H\leq G$, and $H$ is a simple group such that $[G:H]=2.$ Prove: $H$ is the only normal subgroup of $G$ or there is a normal subgroup $K\leq G$, $|K|=2$ such that $G=H \...
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Simplicity of O(p,q)

Let $\operatorname{O}(p,q)$ be the indefinite orthogonal group of signature $(p,q)$ (over $\mathbb{R}$) and $\operatorname{O}_0(p,q)$ its identity component. After doing a bit of research, I am given ...
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2 votes
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Prove that a group of order $540$ is not simple

Prove that a group of order $540$ is not simple In this problem I have done the following steps. Sylow's Theorem gives $n_3=10$ and $n_5=36$. Let $P$ be a Sylow $3$ subgroup and $Q$ a Sylow $5$ ...
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A question in the answer of the following question : Prove that a group of order 72 can't be a simple group

I had the following question: A group of order 72 can't be a simple group. As it was asked and user Dietrich Burde gave links to already asked questions. So, I got to know about : How to prove "...
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2 votes
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Let $H\leq G$ where $H$ is max simple. Prove either $G$ is simple or there exists a minimal normal subgroup $N$ of $G$ such that $G/N$ is simple. [duplicate]

Question: Let $G$ be a finite group and $H$ a maximal, simple subgroup of $G$. Prove that either $G$ is simple or there exists a minimal normal subgroup $N$ of $G$ such that $G/N$ is simple. Thoughts:...
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Prove that there doesn't exists a simple group of order 24 and order 48 . [duplicate]

Edit : SOme people are linking my question to solvability of group of order 8p but I don't want to use the concept of solvability of groups. So, Please don't link it to that. I have been following ...
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  • 1,426
5 votes
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Non-abelian simple groups of odd order less than $10000$.

I am trying to solve problem 6.2.16 from Dummit and Foote, namely Prove there are no non-abelian simple groups of odd order $< 10000$. I did something similar for order $<100$, where I showed ...
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Are there any nonabelian simple groups of order 60? [duplicate]

Prove that there are no nonabelian simple groups of order < 60. I am studying abstract algebra from Artin and trying some questions and got struck on this. Let on the contrary there exists a ...
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Simple group of order $60$ is isomorphic to $A_5$

Prove that any simple group $G$ of order $60$ is isomorphic to $A_5$. I am studying abstract algebra from Artin and studying these assignment from a masters level math course. It gives hint. Use the ...
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0 votes
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$N$ is a simple normal subgroup of a group $G$ and $G/N$ has a composition series...

A question in section Normal and Subnormal Series of Hungerford Algebra. If $N$ is a simple normal subgroup of a group $G$ and $G/N$ has a composition series, then prove that $G$ has a composition ...
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2 votes
1 answer
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Isaacs Algebra : A Graduate Course definition for diagonal subgroup

I actually have a number of related questions. In Problem 7.2, Isaacs defines a subgroup $D$ of $G= M \mathop{\dot{\times}} N$ to be diagonal if $$ D \cap M = 1 = D \cap N \text{ and } DM = G = DN. $$ ...
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Proof that the monster group is simple.

I was reviewing an abstract algebra book, and I get the comment that "the monster group" (the set of matrix of size $196833 \times 196833$) has no normal subgroups,I wanted to know if anyone ...
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