Conjugacy classes and the class equation

Let G act on its underlying set by g·x=gxg inverse. The identity acts trivially, and conjugating first by h and then by g equals conjugating by gh. The orbit of x is its conjugacy class. For permutations, conjugation relabels every entry in each disjoint cycle by g, so cycle lengths are preserved. In the full symmetric group, equal cycle type also suffices for conjugacy: choose a relabeling that sends each cycle of one permutation to a cycle of the other with the same length.

The stabilizer of x under this action is the centralizer C_G(x), the subgroup of elements commuting with x. The center Z(G) consists of elements commuting with every element of G. These concepts are different: the centralizer of one x can be large even if the center is small. An element is central exactly when its conjugacy class has size one. In particular, every element of an abelian group forms its own singleton class; conjugacy supplies no additional grouping in that case.

Choose one representative from each conjugacy class in a finite group. Since orbits partition the underlying set, summing their sizes gives |G|. Separate the singleton classes, which are exactly the center, and express each remaining size as a centralizer index. This is the class equation. Its force comes from combining two descriptions of the same finite set: an additive partition into classes and multiplicative divisibility from subgroup indices. Always include one term per class, rather than one term per noncentral element.

Conjugation also explains a useful normality criterion. A subgroup N is normal exactly when it is a union of conjugacy classes: membership must survive every conjugation. This does not mean every arbitrary union of classes is a subgroup; closure and inverses still need checking. When searching for normal subgroups in a small finite group, their possible orders must both divide |G| and be expressible as one plus the sizes of some nonidentity classes. These necessary restrictions can sharply narrow the search without determining every multiplication rule.

Suppose |G| is a positive power of a prime p. Every centralizer has prime-power order by Lagrange’s theorem. Consequently every noncentral conjugacy class has size a positive power of p larger than one, hence divisible by p. The class equation shows that |Z(G)| is also divisible by p. Since the identity is central, the center cannot be empty; its order is at least p. This is an existence proof from arithmetic, not a procedure for locating a particular nonidentity central element.

The vectors lab offers only a geometric analogy: a planar rotation changes a vector’s coordinates while preserving its magnitude. Adjust angle at fixed magnitude and observe the projections change. Conjugation preserves group-theoretic data such as order, but the lab does not calculate conjugates, centralizers, or classes. In applications, use the actual group operation to compute those sets. Invariants help rule out conjugacy; in a general group, sharing an invariant such as element order does not by itself prove two elements are conjugate.