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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7
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0answers
92 views

Why here “the simple group of order $168$ comes nearly as a counterexample”?

I’m reading an old paper by Saad Adnan and I don’t seem to understand a sentence at the very beginning of this paper. Conjecture.$~~$If the finite group $G$ has exactly $2$ conjugacy classes of ...
2
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43 views

Direct products inside finite simple groups

For a finite group $G$, let $d(G)$ be the largest $k$ such that $G$ admits a subgroup isomorphic to a direct product of $k$ non-trivial groups. I am interested in families $G_n$ of finite simple ...
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1answer
24 views

a group $G$ whose order has exactly two prime divisors is not simple, Burnside theorem?

I have to show that a group $G$ whose order has exactly two prime divisors is not simple. I was thinking to use the Burnside theorem which sais that if $\vert G \vert = p^aq^b$ where $p,q$ are primes ...
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1answer
32 views

A finite group $G$ is called an $N$-group if the normalizer $N_G(P)$ of every non-identity p-subgroup $P$ of $G$ is solvable.

Prove that if $G$ is an $N$-group, then either (i) $G$ is solvable, or (ii) $G$ has a unique minimal normal subgroup $K$, the factor group $G/K$ is solvable, and $K$ is simple. Suppose that $G$ is an $...
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1answer
39 views

Let $G$ be a finite non-solvable group, each of whose proper subgroups is solvable.

Show that $G/\Phi(G)$ is a non-abelian simple group, where $\Phi(G)$ denotes the Frattini subgroup of $G$ So $G/\Phi(G)$ can't be abelian since if it were then is would be solvable and since $\Phi(G)$ ...
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1answer
58 views

A Finite Group, $G$, Contains a Proper Subgroup of Index 2, Thus $G$ is not Simple

Show that if a finite group, $G$, contains a proper subgroup of index $2$ in $G$, then $G$ is not simple. Proof Let $H$ be a proper subgroup of index $2$ in $G$. We know that $H$ is normal because it ...
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1answer
29 views

Finite simple group has order a multiple of 3?

Checking the list of finite simple groups, it seemed to me that all groups have order a multiple of $3$. This clear for alternating groups and checked case by case for sporadic groups. For groups of ...
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1answer
50 views

Irreducible components are the building blocks?

I already heared a lot about: irreducible things are the building blocks everywhere: In groups: the building block are the simple groups. In representation theory: the building blocks are the ...
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1answer
48 views

Normal simple subgroup

Suppose that $S \lhd G$ is a non-abelian simple normal subgroup of $G$. Further, suppose that every automorphism given by the action by conjugation of $G$ on $S$ is an inner automorphism of $S$. Then $...
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1answer
23 views

Let be $A_n$ the alternating group $n\geq 5$. How to prove that $\operatorname{Stab}_{A_n}(x) \cong A_{n-1}$, for all $x \in \{1, …, n\}$?

MY ATTEMPT: I have 3 properties in hand: $H\trianglelefteq G \iff Hg=gH, \; \forall g \in G \iff gHg^{-1}=H, \; \forall g \in G$; When $H=gHg^{-1}, \; \forall g \in G$ then we have $H\...
2
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1answer
63 views

Show there are are no simple groups of order 1638

I'm trying to use Sylow Theory to show there are no simple groups of order 1638. I got so far as to factor $1638 = 2*3^2*7*13$ and compute that we must have $n_2|819$ $n_3 \in \{1,7,3\}$ $n_7 \in \{1,...
16
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1answer
230 views

How do we compute the order of the Monster group?

How do we compute the order of the Monster group? The answer is quoted in many places, but when I trace back the references, I can't find any place where it's computed, or even a sketch of the ...
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1answer
43 views

On the intersections of maximal subgroups of finite simple groups [closed]

I am looking for a proof or a counterexample for the following proposition: Let $G$ be a finite simple group and $M_{1}$ and $M_{2}$ be two maximal subgroups of $G$ with nontrivial intersection.Then ...
2
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3answers
70 views

Let $G$ be a simple group. Show that if $H$ is a subgroup $G$ such that $[G:H]= 3$ then $H=\{1\}$ and $|G|=3$.

MY ANSWER: Since $G$ is simple the only normal subgroups of $G$ are $\{1 \} $ is $ G $ itself. As $ G / H $ is a quotient group it follows that $ |H|\neq 3 $ because otherwise the $ H $ index in $ G $ ...
3
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2answers
108 views

Simple group of order $p^2 q r$, where $p, q, r$ are distinct primes, is isomorphic to $A_5$

As stated, I need to prove that, up to isomorphism, the only simple group of order $p^2 q r$, where $p, q, r$ are distinct primes, is $A_5$ (the alternating group of degree 5). Now I know the ...
6
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58 views

Elementary (CFSG-avoiding) proof that the order of a group of composite order is bounded by the square of the order of its largest proper subgroup?

It seems that the following is true. For a finite group $G,$ define $u(G)$ to be the largest order of a proper subgroup of $G$. Then $\left|G\right|\leq u(G)^2$ provided that the order of $G$ is ...
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0answers
61 views

Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$. [duplicate]

Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$. I feel like I am supposed to use the Index Theorem here but when I use Sylow's Third Theorem I have that $n_{3}\in\{1,4\}$. I am ...
3
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3answers
151 views

Reference request: a list of (small) finite simple groups

I am currently in the midst of a project in which it would be useful to have a list of all (small) simple groups as a means to check calculations, not waste time, verify conjectures for small examples,...
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1answer
52 views

Two questions about non-abelian finite simple groups

In OEIS, I found the positive integers $\ n\ $ , for which there exists a non-abelian simple group with order $\ n\ $ upto $\ 10^{10}\ $. It can be found by entering the numbers $\ 60,168,360\ $ Only ...
4
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1answer
61 views

Does there exist a two-generated simple non-abelian group with specific properties?

Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \...
4
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1answer
37 views

Extensions of $A_5$ by $C_2$.

Recently I've came to result that, if $H$ is a simple group, every homomorphism $\theta :K\rightarrow \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. As an example, I tried to find all ...
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0answers
43 views

Is it possible to take a power series expansion “in the monster group”

Given the monster group, I have no idea what it means to take a power series expansion in this group, I just made up this expression and have been trying to give it a meaningful definition. My main ...
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46 views

Special subgroups of some finite simple groups

Suppose that $G$ is a finite simple group and $d$ a divisor of $|G|$ such that there is no proper subgroup whose index divides $d$. (1) Are there subgroups $H$ and $K$ of $G$ such that $|HK|=d$? (2) ...
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Simple groups satisfying some conditions

We are looking for an example of simple groups $G$ of order $n$ such that the following condition (*) does not hold: (*) for every factorization $n=ab$ there exists a non-trivial subgroup of $G$ ...
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2answers
103 views

If a group $G \cong H$, and if $H$ is a simple group, is $G$ also a simple group? [closed]

I know that is the case, but I asked this question because I cannot find a formal way to prove it. Something is missing in my mind, I don't know what, that's why I asked the question. I don't expect a ...
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1answer
88 views

Asking the proof of maximal normal subgroup in this answer: https://math.stackexchange.com/a/161593/435467

I'm reading the proof of maximal normal subgroup in this answer: https://math.stackexchange.com/a/161593/435467. The question is when $\frac{A}{H} = \frac{G}{H}$, how could we know that $\frac{G}{H}$ ...
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2answers
51 views

A question about Lang's proof of simplicity of $A_n$

Theorem 5.5. If $n\geq 5$ then the alternating group $A_n$ is simple. Proof. Let $N$ be a non-trivial normal subgroup of $A_n$. We prove that $N$ contains some $3$-cycle, whence the theorem ...
4
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1answer
54 views

Representations of simple nonabelian groups

In this post all groups are finite, and all representations are complex linear finite dimensional representations. If a group $G$ is abelian, then all of its irreducible representations are of one ...
6
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0answers
217 views

Is every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ a square element in finite spin group ${\rm Spin}_n^{\epsilon}(q)$?

A. I wonder if every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ is a square element in ${\rm Spin}_n^{\epsilon}(q)$? B. When $Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$, is the generator of $Z(\Omega_{...
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1answer
94 views

A Group of order $p^n$ is not simple [duplicate]

Let $G$ be a group that $|G|=p^n$, with $n \geq 2$ and $p$ prime. Show that G is not simple. I know that, if $G$ is not abelian, then $Z(G) \not= G$ and $Z(G)$ is a normal subgroup of $G$ with $|Z(G)...
9
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2answers
327 views

Nonexistence of a simple group of order 576

Prove that there exists no simple group of order 576. Suppose $G$ is simple of order 576. It is a straightforward application of the index theorem to determine that the number of Sylow 2-subgroups of ...
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0answers
33 views

Centralizer of a Sylow group in the simple group of order 360

Let $P$ be a 3-Sylow group of the simple group $G$ of order 360, which is of order 9. And let $N_G(P)$ be the normalizer of $P$. Then $\frac{N_G(P)}{P}$ acts on $P$. (by conjugation) I want to know ...
8
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1answer
222 views

Nonexistence of a simple group of order 420

From Dummit & Foote, Abstract Algebra, $\S6.2$, Exercise 17(a). Prove that there is no simple group of order 420. Suppose not; label such group $G$. the number of Sylow 7-subgroups of $G$ is ...
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0answers
35 views

How to check whether a given matrix is in the image of a representation?

Let $G$ be a compact simple Lie group, and let $\rho$ be an irreducible representation thereof of $\mathbb K$-dimension $n$, where $\mathbb K=\mathbb C/\mathbb R/\mathbb H$ if $R$ is real/complex/...
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1answer
74 views

A question on characteristically simple groups

Why do non-abelian characteristically simple groups have no nontrivial subnormal subgroups of prime power order? Let $N$ be a finite non-abelian characteristically simple group. A finite group is ...
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1answer
60 views

Question on proof $G$ is not simple

I have a question on the proof of: Proving if $|G|=280$, then $G$ is not simple $n_7=8$ so there are $48$ elements of order $7$. $n_5=56$ so sthere are $224$ elements of order $5$. Now, in the ...
3
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1answer
101 views

Every finite simple group of order $n \geq 3$ is isomorphic to a subgroup of $A_n$

Let $G$ be a finite simple group of order $n$. Prove that if $n \geq 3$, then $G$ is isomorphic to to a subgroup of $A_n$, the alternating subgroup of the symmetric group $S_n$. My idea here was to ...
2
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1answer
46 views

Large abelian subgroups of perfect groups

It is not too hard to prove that if $G$ is a non-abelian finite simple group and $A \leqslant G$ is an abelian subgroup, then $|A| \leq \sqrt{|G|}$ (this can be improved to $\ll |G|^{1/3}$ using the ...
3
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0answers
76 views

Prove that there is no simple group $G$ of order $3^3\cdot7\cdot13\cdot409$

This is from Dummit and Foote Chapter 6.2 Exercise 29 and there is a hint to work in the permutation representation of degree $819$. In Exercise 28 we reduced to: $n_3=2863$ $n_7=47853$ $n_{13}=...
4
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1answer
92 views

Can the fact that $\bigcap _{g\in G}gHg^{-1}\lhd G$ be used to prove that certain groups are not simple?

If $G$ is finite, and $H$ a proper subgroup (which existence is often easy to show by using Sylow Theory), it would be sufficient to prove $core_G(H)=\bigcap _{g\in G}gHg^{-1}\neq \{ 1\}$ to conclude ...
5
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1answer
402 views

Rigorous proof of the Jordan-Hölder Theorem

As far as I am aware, the standard books on abstract algebra (Lang, Dummit, Rotman, Grillet, etc.) do not give a rigorous proof of the Jordan-Hölder Theorem. Here are two examples from Lang and ...
4
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1answer
108 views

Proving that a group of order $p^nq$ for primes $p$ & $q$ is not simple.

Prove that a group of order $p^nq$ for primes $p$ & $q$ is not simple. I've been able to prove the theorem holds for $p=q$ and $p>q$. If $p<q$ the best I've been able to do is use Sylow to ...
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1answer
67 views

Classification of finite simple groups $G$ with $\pi(G)=\{2,3,5\}$ [closed]

Let $G$ be a finite group. The set of prime divisors of $|G|$ is denoted by $\pi(G)$. I am looking for the classification of finite simple groups $G$ with $\pi(G)=\{2,3,5\}$.
4
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1answer
89 views

No simple group of order $2^73^2$

This is a problem from the exam I took this morning "Prove there is no simple group of order $2^73^2.$" This is what I do, suppose there is such a group $G$. Then by Sylow's theorem, $n_2=3,9$. If $...
2
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0answers
32 views

Generalizing CIT-groups to odd case

According to Wikipedia, a $CIT$-group is a group such that the centralizer of any involution is a 2-subgroup. The structure of these groups is known from the works of Suzuki and others. Here is my ...
2
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0answers
62 views

Picture class number-order of the simple groups

Every simple group below are assumed non-abelian. Let us call the class number $k(G)$ of a finite group $G$ the number of its conjugacy classes (also, the number of its irreducible complex ...
3
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61 views

Simple proof of $A_n$ being simple?

I come across an incomplete proof of the simplicity of $A_n,n\geq 5$, in a textbook. The proof looks simple, so I was attracted to it. Here is the proof after I filled in the details. Everything ...
1
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1answer
111 views

How many composition series of group D6 are there? (Dihedral group order 12)

I think I can start from Normal subgroups of D6: $<r>$,$<s,r^2>$, $<sr, r^2>$, and $<s,r^3>$. Because the other normal subgroups $<r^2>$ and $<r^3>$ do not make ...
0
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1answer
24 views

For each $n$ determine whether $S_n$ (1) and $D_n$ (2) are simple or not.

For each n determine whether $S_n$ (1) and $D_n$ (2) are simple or not. So I know that a group is called a simple group if that group has no proper nontrivial ...
4
votes
1answer
49 views

Is there any relationship between Simple Group and Field?

I have studied definition of simple group. I know the fact that concept of normal subgroup is analogous to ideal in ring theory. A field is a ring which do not have any nontrivial ideal. A simple ...

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