Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

composition series and chief series of $p$-group. How to solve the following Problem? Thanks.

Let $G$ be a group of order $p^n$, $p$ prime. Prove every chief factor and every composition factor is of order $p$.

Is every composition series a chief series of $G$?

share|improve this question

2 Answers 2

up vote 0 down vote accepted

Hints:

Let $\,G\;,\;\;|G|=p^n\;,\; p\;$ a prime, be a group, then use that $\,|Z(G)|>1\;$ and a little induction to show that for any $\,0\le k\le n\;,\;\;G\;$ has a normal subgroup of order $\;p^k\;$ . This already answers (almost...) questions (1)-(2) .

As for the last question: take the dihedral group $\;G:=\{s,t\;;\;s^2=t^4=1\;,\;sts=t^3\}\;$ of order 8, and check the series

$$1\le\langle t^2\rangle\le\langle t\rangle\le G\;\ldots$$

share|improve this answer
    
thanks a lot, Prof. DonAntonio –  Ren Shiquan Sep 15 '13 at 7:56

Hint for the first question: If $G$ is a finite $p$-group and $G\neq1$, then $G$ contains a normal subgroup of order $p$.

Hint for the second question: find a counterexample.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.