# Tag Info

Accepted

### What is the intuition behind simple Lie groups?

Why are finite-dimensional simple Lie groups so special? Because they admit a full classification on the level of (complex) finite-dimensional simple Lie algebras by combinatorial data, e.g., root ...
• 131k

### What is the reason that A5 is simple and A4 is not?

In general, $A_{n+1}$ is the group of rotations of the $n$-simplex; explicitly, $A_{n+1}$ acts by permutation matrices on \Delta_n = \{ (x_0, \dots x_n) \in \mathbb{R}^{n+1}_{\ge 0} : \sum_{i=0}^{n+...
• 420k
Accepted

### Is this "coincidence" about representations of the Monster actually a coincidence?

Griess and Smith prove in Griess, Robert L. jun.; Smith, Stephen D., Minimal dimensions for modular representations of the Monster, Commun. Algebra 22, No. 15, 6279-6294 (1994). ZBL0820.20021. that ...

### How are simple groups the building blocks?

Let $G=G_0$ be a finite group. Consider the set of proper nontrivial normal subgroups of $G_0$. If this set is empty, then $G_0$ is simple. Otherwise, the set is ordered by inclusion and we may ...
• 12.2k
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### How do we compute the order of the Monster group?

After more searching, I found a satisfactory answer in the paper "A uniqueness proof for the Monster" by Griess, Meierfrankenfeld, and Segev. The main theorem states: Let $G$ be a finite group ...
• 33.9k

### Sort-of-simple non-Hopfian groups

Here is a nonabelian (countable) example: Let $S$ be a permutation group on a set $X$ with distinguished point. By the wreath product $G\wr S$ I thus mean $G^X\rtimes S$, where $S$ permutes the ...
• 17.9k

### How do I show that every group of order 90 is not simple?

This is a very very late answer. But anyway I'll post for the use of people who are learning form this site (like myself). First note that $90=2\times 3^2\times 5$ and of the form $pq^2r$ with three ...
• 18.3k

• 87.5k

### Rigorous proof of the Jordan-Hölder Theorem

Both invoke Schreier's Theorem (Lang implicitly). Lang's statement of Schreier's Theorem is: Theorem 3.4. (Schreier) Let $G$ be a group. Two normal towers of subgroups ending with the trivial group ...
• 399k
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### Reference request: a list of (small) finite simple groups

Here is a list of orders of nonabelian simple groups up to 10000. Of course, in addition, there is a abelian simple group of order each prime. You can already see from this short list that the most ...
• 90.2k
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### What is the reason that A5 is simple and A4 is not?

Since we're being heuristic anyway, it might be easier to think about $S_n$. The reason that $S_n$ doesn't want to have a lot of normal subgroups is that conjugation in $S_n$ moves stuff around ...
• 30.4k

### Compact simple group which is not a Lie group

kabenyuk's answer is correct but we don't have to appeal to the Gleason-Yamabe theorem. By the Peter-Weyl theorem a compact (Hausdorff) group $G$ has the property that its finite-dimensional unitary ...
• 420k
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### Isaacs Character Theory - exercise 4.11

For Question 1, to show that $t = |G|/q$, it is sufficient to show that all involutions in $G$ are conjugate. There is a hint on how to do that in the book. If not, choose two non-conjugate ...
• 90.2k
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• 90.2k
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### Is every finitely generated simple group $2$-generated?

I haven't read it thoroughly enough to understand the construction, but this 1986 paper by Guba constructs a finitely generated simple group all of whose $2$-generated subgroups are free, and which is ...