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 ...

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2
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2answers
39 views

$G \cong H$ and $G$ is simple. Then $H$ is simple as well.

I think it must be true. Yet I have no rigorous proof for that. So, what I need to prove is that "group being simple" is invariant under isomorphism. That if $G \cong H$, then either both are simple ...
2
votes
2answers
23 views

A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
2
votes
2answers
17 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
0
votes
1answer
66 views

$G$ be a finite simple group , then every element of $G$ can be written as a product of $n$-th powers of elements of $G$?

Let $G$ be a finite simple group , let $n$ be a positive integer such that not all $n$-th powers of elements of $G$ are identity , then is it true that every element of $G$ can be written as a ...
3
votes
1answer
70 views

Questions about $\mathrm{SL}_2(\mathbb{F}_7)$

Let $G=\mathrm{SL}_2(\mathbb{F}_7)$, which has order $336=2^4\cdot 3\cdot 7$. And I may assume that $G$ is generated by the two matrices $$\begin{pmatrix}1&1\\0&1\end{pmatrix}, \begin{pmatrix}...
3
votes
0answers
27 views

Are all simple finite groups unique? [duplicate]

I have studied simple groups of order $60$ and $168$. Specifically, I learned two things below: All simple group of order $60$ are isomorphic to each other. All simple group of order $168$ ...
0
votes
0answers
36 views

Simple subgroups of a symmetric group

In class we used in an exercise that "the only simple subgroup of a symmetric group (if it has one) is the alternating subgroup". But I don't understand where this comes from. Can someone help me?
16
votes
1answer
135 views

Do there exist simple groups of order $n(n+1)$ for some integer $n>1$?

In the comments to this question, it was remarked that the question of whether a finite group of order $p(p+1)$ is necessarily not simple becomes fairly interesting if we do not assume that $p$ is a ...
2
votes
0answers
20 views

Find an isomorphism $PGL_2(F_3) \cong S_4$

I am struggling to find an explanation why this true, I know and I'm sorry that kind of question is commonly asked , although I couldn't find anything about this particular question. Help is ...
1
vote
2answers
16 views

Cycle decomposition of an element of prime order

I am reading the proof of the theorem that every alternating group $A_n$ is simple for $n \ge 5$ in Artin's Algebra. In one step, Artin said We are given that $N$ is a normal subgroup and that it ...
0
votes
1answer
52 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that $12p=2^2\...
1
vote
1answer
52 views

There is no simple group of order $1452$.

I want to prove that there is no simple group of order $1452$. We have $1452 = 2^2\cdot 3\cdot 11^2$, and the Sylow theorems give: \begin{align} n_2 &\in \{1,3,11,33,121,363\} \\ n_3 &\in \{...
1
vote
0answers
67 views

Show the falsity of the statement “all nontrivial finite simple groups have prime order”

This is from exercise 19 in chapter 15 of Fraleigh's book "A first course in abstract algebra". True or false: All nontrivial finite simple groups have prime order The answer to this is false ...
1
vote
1answer
25 views

How to prove a group of order $144$ is not simple using **Normalizers of Sylow intersections**.

How to prove a group of order $144$ is not simple using Normalizers of Sylow intersections. Here's what I have tried, but I am unable to proceed further. And if I proceed with Sylow-$2$ subgroups ...
5
votes
2answers
204 views

Show that the group is not simple

I want to show that: If $G$ contains a subgroup with index at most $4$ and $G$ has not a prime order, then $G$ is not a simple group. Let $N\leq G$ with $[G:N]\leq 4$. We have that $|G|=x\...
1
vote
1answer
24 views

A Group of Length $1$ with a Subgroup of Length $m$

I am trying to find a group $G$ which has a composition series of length $1$, but has a subgroup with a composition series of length $m$ for all $m\geq 2$. I know that length $G=1 \implies G$ is ...
2
votes
1answer
80 views

classify groups of order $36$

Question is to classify all groups of order $36$ I do not even know if it is of my level. Let me try this. Sylow theorem says that there are sylow $2$ subgroups of order $4$ and sylow $3$ subgroups ...
1
vote
1answer
23 views

A lower bound to the index of a subgroup of a non abelian simple group

Let G be a simple nonabelian group and $p$ is the largest prime number which divides $|G|$, prove that if $H$ is a subgroup of $G$ then $|G:H|\ge p$ I tried to show that if $|G:H|<p$ then $H$ is ...
0
votes
1answer
53 views

The group is not simple

I want to show that if $|G|=pqr$ where $p,g,r$ are primes, then $G$ is not simple. We have that a group is simple if it doesn't have any non-trivial normal subgroups, right? $$$$ I have done the ...
0
votes
1answer
21 views

Embedding a simple group of order 60 into $A_6$

I have searched the site for some information on this problem, but the results aren't illuminating enough. Let $G$ be a simple group of order $60$. Since it is simple, then there must be $6$ Sylow $5-...
3
votes
2answers
93 views

How does a short exact sequence say something about a group?

I have a follow-up question to my question here: How are simple groups the building blocks? In that question I asked about what it means when we say that the simple (finite) groups are the building ...
6
votes
1answer
57 views

Which finite simple groups contain $PSL(2,q)$ for some $q\geq 4$?

Which nonabelian finite simple groups contain $PSL(2,q)$ for some $q$? Obviously $PSL(2,q)$ themselves do. Also, as $PSL(2,4)\cong PSL(2,5)\cong A_5\subset A_n,\; n\geq 5$, alternating (nonabelian ...
2
votes
0answers
47 views

Why is $M_{21}\cong PSL_3(\mathbb{F}_4)$?

I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, $...
0
votes
0answers
33 views

How to prove that a matrix lie group is (or is not) simple

I am having some difficulties in proving that some matrix lie groups are or are not simple. I am wondering what I should ask myself to solve this problem quickly. Should I look for normal subgroups? ...
10
votes
2answers
212 views

How are simple groups the building blocks?

I know a bit about simple groups. A finite Abelian group is a (direct) product of finite cyclic groups. The simple finite Abelian groups are exactly $\mathbb{Z}_p$ for $p$ a prime. And so, I ...
1
vote
0answers
40 views

Finite Simple Groups other than $A_n$ and $\rm{PSL}_n$

The finite simple groups taught in undergraduate or graduate courses are only up to $A_n$ or $\rm{PSL}_n$. Even many undergraduate and graduate texts do not consider simple groups beyond these two (...
5
votes
1answer
50 views

If $G$ is a direct product of simple groups, then is every simple subgroup of $G$ isomorphic to a subgroup of some factor?

Let $G=N_1\times N_2\dots \times N_n$. Suppose that $H$ is a simple subgroup of $G$. Is $H$ isomorphic to a subgroup of $N_i$, for some $N_i$? This is a weaker version of this question, which turned ...
-6
votes
1answer
48 views

How to show group with order 125 is not simple? [closed]

How do I show that a group $G$ of order $125$ is not simple? tnx in advance.
1
vote
0answers
31 views

Conjugacy class sizes and classification of finite simple groups

Given a finite group $G$, let $1,n_1,n_2,\cdots, n_k$ denote all the possible sizes of conjugacy classes of $G$, with $1<n_1<n_2\cdots$. The first remarkable theorem by concerning such sequence ...
1
vote
0answers
37 views

What are some good matetials for learning about group extensions?

I'm working through Dummit and Foote's Abstract Algebra for self-study, and I'm interested in reading some more in depth discussions of the group extension problem. Specifically, I am interested in ...
2
votes
1answer
47 views

Is this group simple?

I have this group presentation $G = \langle a,b | ba = a^{-1}b\rangle$. I'm wondering if this group is solvable or not. I tried to show that this group is simple, because if it is, then it can not be ...
0
votes
1answer
47 views

Can a Simple Group possess this property? [closed]

If a simple group G is of order 168 then can I find subgroup of order 7 of G ? If so, then what is the number of subgroups of G of order 7 ?
6
votes
0answers
63 views

Proofs of Simplicity of $A_n$

There are different proofs of simplicity of the group $A_n$, and one can get at least two proofs by choosing randomly 10 books of the subject, so I will not go into what are these proofs? Rather, I ...
9
votes
4answers
155 views

Alternative way to show that a simple group of order $60$ can not have a cyclic subgroup of order $6$

Suppose $G$ is a simple group of order $60$, show that $G$ can not have a subgroup isomorphic to $ \frac {\bf Z}{6 \bf Z}$. Of course, one way to do this is to note that only simple group of order $...
1
vote
2answers
57 views

Subgroup of $S_n$ that has no subgroups of index 2

Let $n\geq5$, $G$ be a subgroup of $S_n$ s.t. $G$ has no subgroups of index 2 (G is also simple) and there is an injection (morphism) from $G$ to $S_n$. Is this enough to say that $G$ lay in $A_n$ ? ...
3
votes
0answers
53 views

Historical notes on the Jordan-Hölder program

I'm looking for any material (books, articles..) documenting the historical process of the formulation, partial work and/or the actual stage of the Jordan-Hölder program. I'm not sure if there is any ...
1
vote
1answer
44 views

Limitations on the structures of normal subgroups and generating a n-degree polynomial formula

I was considering the problem of expressing the roots of a general polynomial $$ a_0 + a_1 x + ... a_n x^n$$ where $a_i, x \in \Bbb{C}$ Roots of course cannot be solely expressed using the field ...
2
votes
1answer
41 views

Simplicity of homeomorphism group

I'm working on the proof of a theorem from A.D. Anderson, about the simplicity of the group of all homeomorphisms of some "sufficiently set-wise homogeneous spaces". Under some conditions on the ...
1
vote
0answers
31 views

Show that there does not exist a simple group of order $120$. [duplicate]

Show that there does not exist a simple group of order $120$. By the Sylow's theorem, I already know that $N_5 | 24$ and $N_5 \equiv 1 \pmod 5$; I found that $N_5 \in \{1,6\}$ I think I can use the ...
2
votes
2answers
59 views

Show that there does not exist a simple group of order $126$.

Show that there does not exist a simple group of order $126$. By the Sylow's theorem, I already know that $N_7 | 24$ and $N_7 \equiv 1 \pmod 7$; I found that $N_7 \in \{1,8\}$ I think I can use the ...
0
votes
0answers
33 views

Importance of centralizer of involutions in finite group theory

I just read the chapter 1 of the book Finite Group Theory by John Rose, entitled "introduction to finite group theory". There, the author introduces the idea of the centralizer of an involution as an ...
1
vote
0answers
43 views

Normal simple subgroups and composition series

The following is an exercise from an algebra course I am taking: Let $G$ be a simple group with composition series $G = G_0 \vartriangleright G_1 \vartriangleright \cdots \vartriangleright G_n = \{1\}...
0
votes
1answer
85 views

Proof of existence of simple group of Order 168 in Dummit and Foote

So I want to show that GL(V) is a simple group, where V is a three-dimensional vector space over the field of 2 elements. I am following Dummit and Foote (last paragraph on p.211) but there is one ...
0
votes
0answers
41 views

$PSL(n,q)$ is simple: Proof

That group $PSL(n,q)=SL(n,q)/(center)$ is simple is proved on the following lines: (1) $SL(n,q)$ acts on (non-zero elements of) $n$-dimensional vector space over $\mathbb{F}_q$. (2) Action of $SL(n,...
0
votes
0answers
36 views

Finite almost simple groups in Testerman-Malle

In Linear Algebraic Groups and Finite Groups of Lie Type by Testerman and Malle, one can read the following on page 249: Let $G=\mathrm{Cl}(V)^F$ be a classical group over the finite field $F$. ...
1
vote
2answers
64 views

A group G with order $15$ is simple?

A group $G$ with order $15$ is simple? There are a theorem for realize it? Thanks for all you help!
-2
votes
1answer
51 views

Subgroup $Q$ of $M\times M'$ is not a direct product of subgroups of $M$ & $M'$, do $M$ and $M'$ share no simple subgroups?

Let $M$ and $M'$ be groups. Let $M\times M'$ be a direct product. If a subgroup $Q$ of $M\times M'$ is not a direct product of subgroups of $M$ and $M'$, in other words, $Q\neq \{(m,m') \mid m\in P\le ...
0
votes
2answers
41 views

No simple subgroups in common implies order of groups relatively prime? [closed]

Given two finite groups A,B. If these two groups share no simple subgroups in common, can we conclude that the orders of these two groups are relatively prime?
0
votes
1answer
18 views

$G\times H$ is not a simple group for $|G|, |H|\geq 2$

I would appreciate if you could please express your opinion about my proof and maybe give me a hint where you deem suitable. Proof: Define a homomorphism as follows $\phi: (G\times H)\to G$ by $\phi[...
3
votes
0answers
50 views

$A_4$ is not simple

I would appreciate if you could please express your opinion about my proof and maybe give me a hint where you deem suitable. To prove that $A_4$, an alternating group of even permutations of $S_4$, ...