# Tagged Questions

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 ...

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 ...
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 $S_n$ ...
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 ...
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 ...
70 views

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 ...
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 ...
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 ...
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 ...
43 views

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 $F$. ...
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!
### $A_4$ 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 $S_4$, ...