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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 ...

3
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1answer
30 views

When a simple group normalizes a subnormal subgroup of a finite group

Out of curiosity I am working my way through Isaacs's Finite Group Theory and am stuck on problem 2A.4: Let $(G,*)$ be a finite group with simple subgroup $N$ such that $\forall H \lhd\lhd G: ...
5
votes
1answer
92 views

How to show that Group of order $2376$ is not simple

How to show that Group of order $2376$ is not simple, Now I know that $2376=2^3.3^3.11$ So, $n_{11}=1,12$(Are there any other possibilities? According to my calculation; these are the all) Now if I ...
2
votes
2answers
38 views

A group of order $p^nm$ isn't simple

Suppose $|G|=p^nm$ with $1<m<p$. Show that $G$ is not simple. I know it has to do with group actions. My idea is to consider a subgroup of order $p^n$, call it $H$. It has index $m$, so there ...
4
votes
1answer
71 views

For odd primes $p$, are finite groups with self-normalizing Sylow $p$-subgroups solvable?

Is it the case that for odd primes $p\geq5$, all finite groups with self-normalizing Sylow $p$-subgroups are solvable? The simple group of order 168 shows that this conjecture does not hold for $p=2$. ...
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1answer
100 views

Can one prove that a group of order 150 is not simple using element counting?

I'd like to know whether it's possible to show that a group of order 150 is not simple using only element counting (or mostly element counting). I have seen a solution to this problem here (Every ...
0
votes
1answer
63 views

A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple. [duplicate]

Let $G$ be a finite group of even order, whose $2$-Sylow subgroups are cyclic. Show that $G$ is not simple. I was trying to use Cayley theoem, place $G$ in some $S_m$ and get a contradiction with $...
2
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2answers
76 views

Simple groups in group theory 2

Problem Let $G$ be a finite group and $H \subset G$ be a subgroup of index $\lvert G:H \rvert =n$. (a) Show that $\lvert H:(H\cap gHg^{-1})\rvert \leq n$ for all $g\in G$ (b) If $H$ is a ...
3
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0answers
85 views

Could every finite simple group be related to a pair of Lie Groups?

In terms of the classification of simple groups, it is known that every finite simple group is either: Cyclic Alternating Of Lie type One of the 26 Sporadic groups On the other hand there are 5 ...
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0answers
24 views

Showing that a group of order 96 is not simple. [duplicate]

I am trying to prove that a group $G$ of order 96 is not simple. My approach is the following: Let $P$ be a Sylow 3 subgroup and let $Q$ be a Sylow 2 subgroup. Then by the Sylow's theorems we know ...
0
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1answer
92 views

Prove that there is no simple group of order $144$

I was reading the following proof for that question (Joanpemos' answer)- How to prove a group of order $144$ is not simple using Normalizers of Sylow intersections. And I understood it well up to ...
3
votes
2answers
72 views

The order of the group $A_n$

I was reading the following proof of why there is no simple group of order $120$: A group of order $120$ cannot be simple And I couldn't understand the following: "so $A_6$ has a subgroup of order $...
2
votes
3answers
115 views

Show that every group of order $5865$ is cyclic

I need help in proving that every group of order $5865$ is cyclic. I thought at the beginning that if I show that every such group is simple and abelian, It will be isomorphic to some $\mathbb{Z}/p\...
2
votes
1answer
51 views

A finite simple group with subgroup for every divisor of $|G|$ is abelian

Let $G$ be a finite simple group. Assume that for every positive integer $d$ that divides $|G$, there is a subgroup $H$ of $G$ such that $|H| = d$. Prove that $G$ is abelian. So I'm really out of ...
1
vote
1answer
68 views

How to prove that there is no simple group of order 10 000 000?

How to prove that there is no simple group of order 10 000 000? Here is what I have so far: Let $G$ be such a group, we have $|G|=10 000 000=2^7*5^7$. To prove that $G$ is not simple a solution ...
0
votes
2answers
59 views

How to show that the intersection of 2 distinct p-Sylows (p prime) is the identity?

Let us suppose that $G$ is a simple group of order 90. Show that 2 distinct 3-Sylows cannot contain the same element $g \neq e_G$ where $e_G$ is the identity. First we can compute the number of 3-...
2
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3answers
102 views

How to prove that a finite group of order 280 is not simple?

Let $G$ be a finite group of order 280. How to prove that $G$ is not simple? A way to do it is to prove that there exists a p-Sylow subgroup of G that is normal, ie that there is a unique p-Sylow ...
2
votes
1answer
45 views

Prove a group of order $p(p+q)$ is not simple for primes $p$ and $q$ where $ p \geq q$.

So my question is the title; I have been self studying for an exam coming up and this problem showed up. I am not sure how to proceed I can't seem to see how the $n_p(G)$ is related to $p+q$. Any hint ...
1
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0answers
43 views

Minimal degree representation of finite simple groups

For this question I am taking a representation of a group $G$ to be a homomorphism from $G$ to $GL_n(K)$ for some field $K$. The degree of the representation is $n$. I am trying to understand the ...
2
votes
1answer
51 views

Groups for which all normal subgroups are perfect

I am trying to understand the following property of groups. A group $G$ is perfect if $G=[G,G]$. So call $G$ extra-perfect if every normal subgroup of $G$ is perfect. One obvious class of extra-...
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0answers
35 views

Conjugacy class with $1$ element in an Infinite Simple Group

We would like to show that if $G$ is an infinite simple group, then the only conjugacy class of exactly one element is $\{1_G\}$. My thoughts: We want to proove that if $|\mathrm{orb}(x)|=1\iff \...
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1answer
61 views

Is non-abelian simple group complete?

Let $G$ be a non-abelian simple group. I wonder if $G$ is complete; i.e., $\mathrm{Inn}\,G = \mathrm{Aut}\,G$. Although I am an elementary learner, I know, just by simple calculation, that $\mathrm{...
0
votes
1answer
29 views

Proving that in a finite abelian group, there exist a chain of subgroups so that the quotient is simple.

If $G$ is an abelian finite group, how can I prove that there exist a chain of subgroups $$\{1\}\subseteq H_1 \subseteq H_2 \subseteq \ldots \subseteq H_n=G$$ So that $H_{i+1}/H_i$ is simple? I don't ...
0
votes
2answers
28 views

Why do the integers modulo a composite positive number never form a simple group under addition?

Let $p,q \in \mathbf{Z}$ such that $p > 1$ and $q > 1$. Then $\mathbf{Z}/pq\mathbf{Z}$ does not form a simple group under addition. Why?
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1answer
60 views

Coprime action of simple groups

Let $A$ act on $G$ coprimely by automorphism where $G$ is a nonabelian simple group. Does it imply that $|A|=p$? (where $p$ is a prime number) If not, Is there any source that examines such cases ...
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votes
1answer
49 views

Are all non-abelian groups not simple?

We know that for all $n\geq 3$ the group $S_n$ is non-abelian. We also know that for all $n\geq 3,$ the group $S_n$ is not simple, because $A_n$ is a normal subgroup of $S_n$ which is not trivial. ...
4
votes
1answer
305 views

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

There are many famous proofs. However, I cannot find a solution using the following method which is also clear. It’s absolutely correct. Has anyone seen this proof before, and do you have a reference? ...
11
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0answers
296 views

Groups of order $180$, $540$, $1080$ are not simple.

Here's how I solve the problems. Thanks for pointing out what might be the weakness of my solutions. Actually, what I want are other ways of solving this kind of problems, appart from counting the ...
1
vote
1answer
43 views

If $G$ is non-abelian and simple then $|G|$ divides $n_p!/2$

If $G$ is non-abelian and simple then $|G|$ divides $n_p!/2$ where $n_p$ is the number of Sylow p-subgroups. I am struggling to prove this. I am considering the conjugation action so I know $G$ must ...
4
votes
0answers
88 views

Finite groups with 15 or 16 conjugacy classes [closed]

How can I classify all almost simple groups with 15 or 16 conjugacy classes? A finite group $G$ is almost simple if there is a non-abelian simple group $S$ such that $S\trianglelefteq G\leq \...
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0answers
24 views

a central extension of $PGL(n,q)$

If $G/Z(G)\cong PGL(n,q)$ and $Z(G)\leq G'\cong SL(n,q)$, then can we conclude that $G\leq GL(n,q)$? If yes, what is the reason?
1
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1answer
132 views

Center of Simple Abelian Group and Simple Nonabelian Group

I just read about short topic about simple group and I found problem about center of simple abelian group and nonabelian From definition of simple, it must have no proper non trivial normal subgroup....
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1answer
107 views

Every finitely generated group has simple quotient

I've read that every finitely generated group has a simple quotient. Is it obvious?
18
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1answer
315 views

Discovery of the first Janko Group

Recently, I was reading about Janko's discovery of $J_1$, the first “modern” sporadic simple group. Janko and others were trying to classify all finite simple groups with an involution centralizer ...
6
votes
1answer
86 views

Prove that there is no subgroup of index $6$ in a simple group of order $240$

Let $G$ be a group of order $240=2^4\cdot 3\cdot 5$. Assume $G$ is simple, then show that there is no subgroup of index 2, 3, 4 or 5 show that there is no subgroup of index 6 For ...
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0answers
66 views

A group of order 189 is not simple

I seek to prove that a group G of order 189 is not simple. So, for contradiction, I assume G is simple. $|G|=189=3^3 7$. Now, by the Sylow theorems, $n(7)=1+7k$ divides $3^3=27$. But this is only ...
0
votes
0answers
22 views

simplicity of PSU group

using Iwasawa's lemma, prove the simplicity of projective special unitary group $PSU(n,q)$. I have not found googling a book/reference which contain the proof, only found the proof simplicity of $PSL$...
4
votes
2answers
71 views

If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple.

If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple. By the Sylow theorem, we have that the number of $2$-sylow subgroups of $G$ $n_2$ satisfy that $$ n_2 \equiv1\mod2\mbox{ ...
2
votes
1answer
115 views

Show that the semi-group of matrices $\begin{bmatrix}x&0\\ y&1\end{bmatrix}$ with $x>0$ and $y>0$, is simple

Let $S=\left\{\begin{bmatrix}x&0\\ y&1\end{bmatrix}: x,y\in (0,\infty)\right\}$ be a semigroup under matrix multiplication. How to prove $S$ is simple but neither left nor right simple? the ...
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1answer
23 views

Graph Theory Question! [closed]

Give an example of a simple graph G such that for each pair of distinct vertices a and b, there are exactly 2 paths from a to b?
3
votes
1answer
96 views

Prove that a group of order 2000 or 4000 is never simple

Suppose that $G$ is a group with order $2000$. Prove that the group is not simple. How about when the group order is $4000$? Here is my work so far, for the order $2000$ case: Note that $2000 = 2^45^...
1
vote
1answer
28 views

If $ G$ is an almost simple finite group, then $G$ has no nontrivial normal abelian Sylow subgroup

Why the almost simple finite groups has no nontrivial normal abelian Sylow subgroups? Any illustrations or recommended books to understand this idea?
11
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1answer
162 views

Characterization of $A_5$ by the Centralizer of an Involution

In his thesis, Fowler had shown that $A_5$ is the only finite simple group $G$ affording an involution $u$ such that $C_G(u) \cong C_2 \times C_2$. Is there a proof of that result that relies on ...
0
votes
1answer
23 views

Why they doesn't consider intersection of perticular order?

I was reading this, question (shown in pic). But I didn't get, in (3) why they doesn't consider case of $|Q_i ∩ Q_j| = 9$ ? Since, $Q_i ∩ Q_j ≤ Q_i$ $→ |Q_i ∩ Q_j| = 9$ is also valid case and if $|...
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0answers
22 views

A problem to understand the given exercise

I am trying to solve a problem which says,Let $G$ be a group and $H$ be a subgroup of finite index.show that $G$ contains a normal subgroup of finite index. Actually I cann't except this statement as ...
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0answers
26 views

Simple groups and the number of sylow p-subgroups

Our professor assigned us the following T/F question: A simple group has more than one Sylow p-subgroup for every prime p. His answer was: No, but yes if the prime p divides the order of the group. ...
0
votes
1answer
47 views

Conflict with finite Moufang loop solvability propositions

I really got stuck with the following contradiction. Say we have a Moufang loop $Q$, $|Q| < \infty$. To put it briefly, Moufang loops are groups that not necessary be associative, with extra ...
2
votes
1answer
99 views

GAP for Design Theory

In journal titled "Unitals, Projective Planes and Other Combinatorial Structures Constructed from the Unitary Groups U(3,q), q=3,4,5,7", in the picture attached, GAP used there, the word: "Using ...
1
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1answer
39 views

group theory simple group subgroup index |G| < n!

Show that if $G$ is a simple group with a subgroup $H$ of index $n>1$, then $|G| \leq n!$. Hence show that a group of order $2^k \times 3$ can never be simple for $k>1$. So I have let $X$ be ...
8
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0answers
66 views

An upper bound on automorphism orbit lengths in nonabelian finite simple groups

In what follows, for a group $G$, $\operatorname{Aut}(G)$ and $\operatorname{Out}(G)$ denote the automorphism and outer automorphism group of $G$ respectively, and for $x\in G$, $x^G$ and $x^{\...
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votes
0answers
24 views

About the degree of character of $PSL(n,q)$.

It is well known that for $n\geq2$ the group $PSL(n,q)$ is simple except for $PSL(2,2)=S_3$ and $PSL(2,3)=A_4$. Let $G$ be one of the simple groups $PSL(n,q)$. From the ATLAS of finite group, we ...