Is that truth that any group of order 224 is abelian?

Question

I'm trying to solve this problem which is connected with Group Theory.

As I've already noticed any group of order 224 must be non-simple. How then I should check if it is abelian or not (in general case)?

If a group has even order (greater than $4$) then it need not be abelian (think dihedral). — Tobias Kildetoft, Dec 14 '16 at 09:46
Yeah, you're right. But in this case we know additionally that it is not simple. — JackieWilshere, Dec 14 '16 at 09:48

score 1 · Accepted Answer · answered Dec 14 '16 at 09:54

1

Every group of order $n$ is abelian iff $n$ is a cubefree nilpotent number.

Therefore, not all groups of $224$ can be abelian because $224=2^5 \cdot 7$ is not cubefree.

(You don't even need to known what a nilpotent number is.)

answered Dec 14 '16 at 09:54

lhf

216,483

See http://math.stackexchange.com/a/156492/589 and http://math.stackexchange.com/questions/227628/what-are-the-situations-in-which-any-group-of-order-n-is-abelian – lhf Dec 14 '16 at 10:02

score 1 · Answer 2 · answered Aug 17 '23 at 19:01

The dihedral group $D_{112}$ has order $224$ and is not abelian. So it is not true.

We also know that every group of order $224$ is solvable, but this doesn't matter here. Indeed, all dihedral groups are solvable. We only need to find one non-abelian group of order $224$ to refute the claim that all groups of order $224$ are abelian.

Is that truth that any group of order 224 is abelian?

2 Answers2