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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 $G$?

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up vote 1 down vote accepted


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

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

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thanks a lot, Prof. DonAntonio – Phosphorus 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.

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