2

EDIT: Let $G$ be a finite group with a unique minimal subgroup. Then we know that $G$ is a $p$-group. Let $p\neq2.$

Is it true that every subgroup $H\le G$ and every quotient $G/N$ have a unique minimal subgroup?

Thanks

Joe
  • 12,091

1 Answers1

3

No. The quaternion group of order $8$ is a counter-example, which has the Klein $4$-group as a quotient. However, I think it is true if the order of the group is odd since, in that case, the group is cyclic of order equal to a power of an odd prime number.

James
  • 9,442
  • Yeah you're right. I missed an important hypotesis. I'm going to edit. – Joe Sep 19 '14 at 22:13
  • My problem is that the conclusion "the group is cyclic" is the one I'm searching for. Hence I can't use it! The question I posted, is useful in the induction part of the proof of this theorem. – Joe Sep 19 '14 at 22:30
  • @Joe: See this question: http://math.stackexchange.com/questions/879196/a-persisting-element-in-all-subgroups – Minimus Heximus Sep 19 '14 at 22:44