# Tag Info

### Conjectures for finite groups that fail with large counterexamples

According to this answer by Chain Markov on the post I linked to, here are a few, ordered from smallest to largest (known) counterexample: Automorphism group of a non-abelian group is non-abelian: ...
• 3,247
Accepted

### Test if a finite group is a symmetric group algorithmically

The goal should be to do it in time $O(n!^2)$, which is the size of the input (the multiplication table). First determine the partition of $G$ into conjugacy classes. This can be done efficiently with ...
• 11.2k
Accepted

### A modification of a cyclic group that seems to break being a group. What is it?

I will let $a\oplus b$ denote $N-\text{mod}(a+b,N)$. This operation is not even associative, as $$(s\oplus0)\oplus0=s\ne N-s=s\oplus0=s\oplus(0\oplus0)$$ Actually, it is a commutative quasigroup, ...
• 3,156
Accepted

• 40.4k

### Maximal subgroup of a finite group contains all other subgroups.

I want to prove that all other proper subgroups of $G$ except $M$ are contained in $M$. No, it is false. Counter-example, Klein 4 group. $G=\{e, a,b,c \}$, where $\{e,a \}$, $\{e,b\}$, $\{e,c\}$ are ...
• 21.1k
Accepted

### Finite non abelian simple groups whose order is not divisible by 8

There are infinitely many: the groups ${\rm PSL}(2,q)$ with $q \equiv \pm 3 \bmod 8$.
• 90.2k
Accepted

### Understand proof of lemma about existence/uniqueness of group generated by a set

The definition says that the subgroup generated by the subset $S$ is a subgroup with certain properties. But just making that statement does not guarantee that such a subgroup exists. So really before ...
• 90.2k
Accepted

### Does $S_n$ always embed into $GL_{n-1} (\mathbb{F}_p$)?

Okay, I was being a little flippant. There are actually some nontrivial details to check here because of positive characteristic issues so let's check them. Let $F$ be any field. $S_n$ acts faithfully ...
• 420k