- #1

Mr Davis 97

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## Homework Statement

Let ##G## be a finite p-group and ##Z(G)## its center. If ##N \not = \{e\}## is a normal subgroup of ##G##, prove that ##N\cap Z(G) \not = \{e\}##.

## Homework Equations

## The Attempt at a Solution

Since ##N## is a normal subgroup we can let ##G## act on ##N## by conjugation. In a manner similar to the case when ##G## acts on itself, we can construct the following class equation. Let ##n_1,\dots,n_r## be representatives of the orbits of this action not contained in ##N\cap Z(G)##. Then $$|G| = |N\cap Z(G)| + \sum_{i=1}^{r}[G : \operatorname{Stab}_G(n_i)].$$ Since some prime ##p## divides ##|G|## and ##[G : \operatorname{Stab}_G(n_i)]## for all ##i\in [1,r]##, it follows that ##p## divides ##|N\cap Z(G)|##. Hence ##N \cap Z(G) \not = \{e\}##. QED

My main question is have I explained in enough depth how I obtain the class equation that I got? Do I need to show in a rigorous way that $$|G| = |N\cap Z(G)| + \sum_{i=1}^{r}[G : \operatorname{Stab}_G(n_i)],$$ or is what I have written sufficient?