$𝑝$ -group

Given a Prime number $𝑝$ , a $𝑝$ -group $𝐺$ is a group in which the order of every element is an $ℕ 0$ power of $𝑝$ , #m/def/group i.e. for all $𝑥 \in 𝐺$

| 𝑥 | = 𝑝 𝑛

for some $𝑛 \in ℕ 0$ . By Cauchy's order theorem, for a finite group this is equivalent to the order of $𝐺$ being an $ℕ 0$ -power of $𝑝$ , i.e.

| 𝐺 | = 𝑝 𝑛

for some $𝑛 \in ℕ 0$ .¹

Properties

A nontrivial normal subgroup of a finite $𝑝$ -group always has a nontrivial intersection with the centre.²

Proof of 1.

Consider the action of $𝐺$ on $𝑁 ⊴ 𝐺$ by conjugation. The orbits of size 1 are the elements of $Z (𝐺) \cap 𝑁$ . By the Orbit-stabilizer theorem, the size of orbits divide $| 𝐺 |$ , hence all orbits have size $𝑝 𝑛$ for some $𝑛 \in ℕ 0$ . On the other hand, the ground, $| 𝑁 |$ divides $| 𝐺 |$ since the order of a subgroup divides the order of a group. Since $𝑁$ is nontrivial, $| 𝑁 | = 𝑝 𝑚$ for some $𝑚 \in ℕ$ . Now adding the sizes of orbits,

| 𝑁 | = 1 + \dots ⏟ s i z e 1 + 𝑎 1 𝑝 + 𝑎 2 𝑝 2 + \dots = 𝑝 𝑛

so there must be at least one non-identity orbit of size 1, i.e. at least one other central element.

𝑝-group

Properties

See also

Footnotes

$𝑝$ -group