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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41 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, ...
1
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0answers
11 views
+100

Alternative construction of the twisted group $^2 E_6 (q)$.

I am looking for the alternative construction of the twisted finite simple group $^2 E_6 (q)$ possibly avoiding the Lie theory. Especially I am interested in calculating its order. Maybe some of you ...
0
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0answers
27 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? ...
9
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2answers
179 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 ...
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0answers
37 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
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1answer
44 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 ...
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1answer
36 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.
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0answers
25 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 ...
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0answers
33 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
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1answer
42 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
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1answer
46 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
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0answers
49 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
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4answers
85 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 ...
1
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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
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0answers
47 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
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1answer
38 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
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1answer
40 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
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0answers
30 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 ...
2
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2answers
48 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
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0answers
25 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 ...
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0answers
39 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 = ...
0
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1answer
59 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
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0answers
30 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 ...
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0answers
32 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 $F$. ...
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2answers
61 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!
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1answer
49 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
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2answers
40 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
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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 ...
3
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0answers
42 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 $S_4$, ...
0
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0answers
46 views

Why does $A_\infty=\bigcup_{n\geq 5}^{\infty} \,A_n$?

My question is: why does $$A_\infty=\bigcup_{n\geq 5}^{\infty} \,A_n \,\,?$$ Doesn't $A_{\infty}$ contain (an isomorphic copy of) $A_4$, which is not a simple group? I'm sorry if this is a stupid ...
1
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1answer
60 views

A question about the composition series of two particular isomorphic groups

The problem is stated as this: "Prove that if $$G:=G_1\times G_2 \times \dots\times G_s \cong H_1 \times H_2 \times \dots\times H_t=:H,$$ where each $G_i$ and $H_i$ is finite simple group, then ...
9
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1answer
184 views

$G$ be a finite simple group and $H,K$ be subgroups of prime index ; then is it true that $H,K$ are of same size?

Let $G$ be a finite simple group and $H,K$ be subgroups of prime index ; then is it true that $|H|=|K|$ ?
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2answers
70 views

Non-commutative simple group operates non-trivially on a set with less than $5$ elements

Either prove or disprove that a non-commutative simple group can or cannot operate non-trivially on a set with less than $5$ elements. What does the term "operate on a set" mean? And also ...
5
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1answer
42 views

Can a non-trivial conjugacy class in $A_n$ contains $<n$ elements?

In one of the proofs of simplicity of $A_n$ ($n\geq 5$), a fact used is the following: There is no (non-identity) conjugacy class in $A_n$ containing $<n$ elements. I initially was using ...
3
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1answer
32 views

A nonsplit extension of a nonabelian finite simple group by a cyclic group of odd prime order

Let $p$ be an odd prime. Does a nonabelian finite simple group $S$ exist such that $H^2(S, \mathbb{Z}/p\mathbb{Z})$ is not trivial?
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0answers
36 views

Nonabelian and simple subnormal subgroups of a group.

Let $S \triangleleft \triangleleft$ $G$, where $S$ is nonabelian and simple. Show that $S^{G}$ is a minimal normal subgroup of $G$. Notation: $S \triangleleft \triangleleft$ $G$ means $S$ is ...
2
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2answers
64 views

Problem from “The Theory of Finite Groups” by Kurzweil and Stellmacher

I'm currently working my way through The Theory of Finite Groups by Kurzweil and Stellmacher and came across the following question in the first chapter: Let $G$ be simple, $|G| \ne 2$, and $f : G ...
0
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0answers
9 views

simple groups of lie type

Let G be a finite simple group of lie type. I want to know is G a finite group of lie type? Also if yes, when type of G is X as a finite simple group of lie type, has G same type as a finite group of ...
2
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2answers
286 views

A group homomorphism from a simple group is injective

Let $G_1$ be a simple group, that is the only normal subgroups of $G_1$ are itself and the trivial subgroup. If $\phi : G_1 \rightarrow G_2$ is a group homomorphism, does that mean $\phi$ is ...
5
votes
1answer
79 views

If $|G|= 2^{3}3^{3}11$, then $G$ is not simple

I have this problem in my notes: If $|G|=2^{3}3^{3}11$, then $G$ is not simple The instructor solved it in a way that I could not follow. The solution I have is attached below. If someone could ...
4
votes
2answers
63 views

for prime $q\ge p$, integer $m\ge 0$, any group with order $p^2 q^m$ is not simple

I tried the two ways, but both are all failed. First, counting the order of union of all Sylow $p$, $q$-subgroup. Second, group action from orginal group to set of cosets by Sylow subgroup. Is there ...
9
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0answers
25 views

Nodes of the Dynkin diagram for even-dimensional orthogonal groups

I'm reading Wilson's book "The Finite Simple Groups", specifically the sections on the orthogonal groups. In section 3.7.4, he discusses subgroups of the orthogonal groups. The Dynkin diagram for the ...
0
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1answer
83 views

Why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$? [closed]

Can anyone please clarify me on why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$ for some positive constant $k$? Thanks
4
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1answer
54 views

Showing group of order 9555 is not simple by using following Hint

There is already another proof of this theorem, but I'm curious about solving this problem as written in Dummit&Foote(Ex.6.2 12) Show there is no simple group $G$ with $|G|=9555$. No simple ...
7
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1answer
69 views

If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple

I'm looking for verification: My claim: If $G$ is a finite group and $H$ is a (proper)subgroup of index $k>1$, where $k! \leq |G|$, then $G$ is not simple. Proof: Consider the set of left cosets ...
1
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1answer
28 views

Thompson sporadic group presentation and conjugacy class representatives

So if you look here: http://web.mat.bham.ac.uk/atlas/v2.0/spor/Th/ they provide matrices, $a$ and $b$, which generate the Thompson sporadic group. They also give a representative for each conjugacy ...
6
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2answers
81 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...
1
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3answers
86 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
3
votes
2answers
120 views

How did we classify the finite simple groups when we haven't classified the primes?

Why was the classification of the finite non-abelian simple groups "easier" (!!) than the classification of the finite abelian simple groups [the prime numbers], which still doesn't exist? (Does it? ...
1
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2answers
61 views

Any simple group of order 60 is isomorphic to $A_5$

Any simple group of order 60 is isomorphic to $A_5$ Let $G$ be a simple group of order $60$ .(Assumption) $G$ has a subgroup of order 12 say $H$ . Then by Extended Cayley's Theorem $\exists ...