Questions tagged [simple-groups]

Question 1

When looking at proofs for why no group $G$ of some order is simple, one frequently mentioned trick is to define a homomorphism from the group to $S_n$, and show that the kernel is a non-trivial ...

Question 2

Let $A \subset \Bbb N$ be the set of natural numbers for which there is a nonsolvable group of order $n$; this set is enumerated in OEIS A056866. What is the natural density $$a := \lim_{n \to \infty}...

Question 3

There is a theorem by Wang and Chen that says: when the finite group $A$ acts via automorphisms on the finite group $G$ with $|A|$ and $|G|$ coprime, and $C_G(A)$ is either odd-order or nilpotent, ...

Question 4

I’m interested in transfers in group theory. It was I. Schur who first introduced the notion of a “transfer homomorphism” to study the simplicity of a group $G$. Let $G$ be a finite group and $H \leq ...

Question 5

It is known that there exists an isomorphism $\mathfrak{su}(2)\oplus\mathfrak{sl}(2,\mathbb{R})\cong\mathfrak{so}^\ast(4)$. Is there some surjection from $\mathrm{SU}(2)\times\mathrm{SL}(2,\mathbb{R})$...

Question 6

I know that the following simple groups have 2-transitive actions: The alternating groups $A_n$ for $n\geq5$ The linear groups $L_n(q)$, except $L_2(2)$ and $L_2(3)$ The unitary groups $U_3(q)$, ...

Question 7

Is there any classification of finite non simple, non solvable group whose all non trivial normal subgroups are non solvable. I am aware of $S_n$ for $n \geq 5$. Is there any other such groups. More ...

Question 8

I've been reading up about character theory of finite groups and one result I see popping up a lot is that, if $S$ is a nonabelian simple group, then elements of $\operatorname{Aut}(S^n)$ act as ...

Question 9

I'm looking for an example of a nontrivial group $G$ with the property that for all nontrivial proper subgroups $H$ of $G$, there does not exist a nonidentity automorphism $\alpha:G\to G$ such that $\...

Question 10

I'm trying to prove that no group of order $p^2 q$ ($p,q$ distinct prime numbers) is simple. Is my idea correct? Call $\ell_p$ the number of $p$-Sylows in $G$. By the Third Sylow Theorem, the number $\...

Question 11

If a group of order $168$ has more than one sylow $7$ subgroups then is it simple? I am relatively new to solving problems like this. I tried solving this problem as follows: Let us assume that $G$ is ...

Question 12

I realise that this question has been posted before here, but while trying to solve it myself, I ran into a block and I'm not sure how to proceed. This is how the question has been laid out in a ...

Question 13

Can a finite non-abelian simple group be expressed as a product of two it's maximal subgroups with trivial intersection? I think it's not possible. The heuristic behind the thought is that if such a ...

Question 14

In "Finite Simple Groups" by Wilson, on page 30, he described the construction of the triple cover of $A_{7}$. He added 18 images of $(2,0,0,0,0,0)$ to the set of 45 vectors which was ...

Question 15

Definition 1: A group $G$ is quasisimple if it is perfect (i.e., $G=[G,G]$, its derived subgroup) and $\operatorname{Inn}(G)$ is simple. NB: I know that $\operatorname{Inn}(G)\cong G/Z(G)$. ...

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Questions tagged [simple-groups]

How do conjugates allow us to make a homomorphism?

Natural density of orders of nonsolvable groups

Wang-Chen theorem on solvability?

Intuition behind the transfer homomorphism

Sporadic isogeny between $\mathrm{SO}^\ast(4)$ and $\mathrm{SU}(2)\times\mathrm{SL}(2,\mathbb{R})$

Can someone give a full list of non-solvable groups with 2-transitive actions?

Non solvable groups with all normal subgroups non solvable.

Automorphism group of a product of simple groups

Example of group such that no nontrivial automorphism fixes a proper nontrivial subgroup? [closed]

No group of order $p^2 q$ is simple [duplicate]

If a group of order $168$ has more than one sylow $7$ subgroups then is it simple?

Let G be the direct product of two nonabelian simple groups. Then G has exactly four normal subgroups. [closed]

Finite simple groups as product of maximal subgroups

Construction of the triple cover of $A_{7}$ in "Finite Simple Groups" by Wilson.

Can the direct product of two quasisimple groups not be one-headed?

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