Group Actions and Sylow Theorems - Key Properties

The class equation and Burnside's lemma are powerful counting tools derived from group actions, with applications throughout combinatorics and group theory.

TheoremClass Equation

Let $G$ be a finite group acting on itself by conjugation. Let $Z(G)$ be the center and $C_1, \ldots, C_k$ be the non-trivial conjugacy classes. Then: $|G| = |Z(G)| + \sum_{i=1}^{k} [G : C_G(g_i)]$

where $g_i$ is a representative of $C_i$ and $C_G(g_i)$ is the centralizer of $g_i$ .

The class equation expresses the order of $G$ as a sum of the center size plus the sizes of conjugacy classes. Each conjugacy class size equals the index of the centralizer, by the orbit-stabilizer theorem.

ExampleApplication to p-Groups

A finite group $G$ is a $p$ -group if $|G| = p^n$ for some prime $p$ . For non-trivial $p$ -groups, the class equation implies: $p^n = |Z(G)| + \sum_{i} [G : C_G(g_i)]$

Since each index $[G : C_G(g_i)]$ is a power of $p$ greater than 1, we have $|Z(G)| \equiv 0 \pmod{p}$ . Thus: $|Z(G)| \geq p$

Every non-trivial $p$ -group has a non-trivial center!

TheoremBurnside's Lemma

Let $G$ be a finite group acting on a finite set $X$ . The number of orbits equals: $\frac{1}{|G|} \sum_{g \in G} |X^g|$

where $X^g = \{x \in X : g \cdot x = x\}$ is the set of elements fixed by $g$ .

Burnside's lemma is invaluable for counting problems with symmetry. Instead of counting orbits directly, we count fixed points for each group element and average.

ExampleNecklace Problem

How many distinct necklaces can be made with $n$ beads, each colored red or blue, where two necklaces are the same if one can be rotated to match the other?

The cyclic group $C_n$ acts on the set of $2^n$ colorings. A rotation by $k$ positions fixes a coloring if and only if the coloring has period dividing $\gcd(n, k)$ . The number of such colorings is $2^{\gcd(n,k)}$ .

By Burnside's lemma: $\text{Number of necklaces} = \frac{1}{n} \sum_{k=0}^{n-1} 2^{\gcd(n,k)}$

Remark

The class equation and Burnside's lemma both stem from the orbit-counting theorem: summing orbit sizes over all orbits equals summing stabilizer sizes divided by $|G|$ . These results transform difficult counting problems into manageable calculations.

Group actions convert abstract group theory into concrete counting tools, bridging algebra and combinatorics.