Articles of simple groups

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple?

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple. However it only lists projective groups in its list of simple Lie groups.

Infinitely many simple groups with conditions on order?

If $G$ is a non abelian finite simple group, we know that $4$ divides $|G|$. More precisely there are infinitely many finite simple groups $G$ such that $v_2(|G|)=2$, just consider $\mathrm{PSL}_2(\mathbb F_p)$ with $p\equiv 3 \pmod 8$, $p$ prime and $p>3$. So my question tries to generalize the above statement: Do we know the primes […]

Why is the Monster group the largest sporadic finite simple group?

I do know that there is a “long proof” (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other bigger group out there? Beyond that long proof, is there any alternative “proof” of why the […]

No group of order $400$ is simple – clarification

I was reading through a proof that no group of order $400$ is simple which can be found here: Here is an outline for a solution. First of all, $|G| = 400 = 2^4 \cdot 5^2\ $. By Sylow’s theorem we know that the number of Sylow 5-subgroups must be a divisor of $2^4$ […]

Hint on an exercise of Mathieu groups

I am working through an abstract algebra exercise book for my exam. It has solutions, but sometimes only references are given to books which I may not have access to. So I think, that the exercise is a bit more advanced and not suitable for an exam. Can somebody tell me, if the following exercise […]

Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) of a simple group. Suppose we have a copy of $\mathbb{Z}_8$ in a non abelian simple group $G$ then $C_G(\mathbb{Z}_8)$ would be of order at least $8$ as $\mathbb{Z}_8$ is […]

Proving that infinite union of simple groups is also simple group

Given that: $$G_1\subseteq G_2\subseteq G_3\subseteq \dots\subseteq G_n \subseteq G_{n+1}\subseteq \cdots$$ Are all simple groups. Prove that $$G=\bigcup_{n=1}^{\infty}G_n$$ is also a simple group. So first of all, I proved that $G$ is a group, there was’nt too much work to do there, but how can I prove that $G$ is simple? thank you in advance.

Simple group of order $660$ is isomorphic to a subgroup of $A_{12}$

Prove that the simple group of order $660$ is isomorphic to a subgroup of the alternating group of degree $12$. I have managed to show that it must be isomorphic to a subgroup of $S_{12}$ (through a group action on the set of Sylow $11$-subgroups). Any suggestions are appreciated!

Centralizer of involutions in simple groups.

I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field, How many conjugacy classes of involution does it have? If there are more than one, what is their sizes? I would be […]

Criterion for being a simple group

In this work it’s written that A group $G$ is simple if and only if the diagonal subgroup of $G \times G$ is a maximal subgroup. How can I prove it?