Determination of conjugacy class structure of symmetric group:S3

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This page describes the process used for the determination of specific information related to a particular group. The information type is conjugacy class structure and the group is symmetric group:S3.
View all pages that describe how to determine conjugacy class structure of particular groups | View all specific information about symmetric group:S3

This page describes various ways to determine the conjugacy class structure of symmetric group:S3. We will consider the following approaches:

  • A completely naive approach based on brute force computation starting from the multiplication table.
  • An approach that uses the fact that cycle type determines conjugacy class, but using naive methods to list all the permutations of a given cycle type.
  • A sophisticated generalizable approach that identifies the conjugacy classes and lists the elements in them.

We also discuss hacks to determine the conjugacy class sizes without actually working with the group at all.

Finding the conjugacy class sizes using number-theoretic constraints

  • We know that the identity element forms a conjugacy class of size 1. This leaves 5 more elements to be partitioned into conjugacy classes.
  • We know that size of conjugacy class divides order of group. This forces that the remaining conjugacy classes have size 1, 2, 3, or 6. Size 6 is ruled out by size considerations. The conjugacy classes of size 1 form the center, which is a subgroup, so by Lagrange's theorem, the number of such conjugacy classes also divides 6, therefore it is 1, 2, 3, or 6.
  • From a purely arithmetic perspective, the following possibilities exist for conjugacy class sizes:
    1. size 1 (6 times)
    2. size 1 (3 times), size 3 (1 time)
    3. size 1 (2 times), size 2 (2 times)
    4. size 1 (1 time), size 2 (1 time), size 3 (1 time)
  • Possibility (1) is ruled out because the group is non-abelian. We can rule out possibilities (2) and (3) in either of two ways: either we use the fact that symmetric groups are centerless, or we use the fact that cyclic over central implies abelian, so the center cannot be a maximal subgroup (and in particular, cannot have order 2 or 3). This leaves possibility (4): the conjugacy class sizes are 1, 2, 3.