We can prove that for any finite group of order n, it must be

>>8148416
Obviously not you cretin, that'd imply all groups are cyclic.

>>8148416
the strongest result in this sense is Cauchy theorem

The group [math] (\mathbb{Z}_2)^3 [\math] is an abelian group of order 8 with no elements of order higher than 2.

>>8148461
Forward slash next time friendo

>>8148457
>>8148461
Thanks, family.

>>8148446
Wow, rude. Does saying that make you feel better about yourself?

>>8148416
You'd be interested in the Sylow theorems.

>>8148748
Are the Sylow theorems supposed to imply anything about general divisors of the order of the group? As far as I know it only tells you something about divisors that are maximal prime powers?

>>8148575
The reason you "couldn't quite get there" is because if you actually used your brain for half a second, you'd see that it's trivially false. Oh sorry, have you not learned about cyclic groups yet?

>>8148798
It's a special case of what you were asking, since the general case won't work.

>>8148913
>>8148798
Of course, I mean to say the corollary of them which states that for any prime p dividing the order of a group, you can find an element of order p.