Group Theory I.4 (Conjugate Permutations)

Let \sigma, \tau \in S_n be any permutations. The element \tau \sigma \tau^{-1} \in S_n is called a conjugate of \sigma. [ Note: for brevity of notation, we've omitted the composition symbol \circ. ]

Let us give a more descriptive formulation of this. We trace the elements 1, 2, …, n through the permutation \tau \sigma \tau^{-1}:

  • first, \tau^{-1} takes \tau(i) to i;
  • next, \sigma takes i to \sigma(i);
  • finally, \tau takes \sigma(i) to \tau(\sigma(i)).

In short, the conjugate permutation takes \tau(i) \mapsto \tau(\sigma(i)), i.e. if we write \sigma as a product of disjoint cycles (see above),

\sigma = (a_1, a_2, \ldots, a_r) (b_1, b_2, \ldots, b_s) (c_1, c_2, \ldots, c_t) \ldots

then the conjugate permutation \tau \sigma \tau^{-1} is obtained by replacing each occurring number k by \tau(k):

\tau \sigma \tau^{-1} = (\tau(a_1), \ldots, \tau(a_r)) (\tau(b_1), \ldots, \tau(b_s)) (\tau(c_1), \ldots, \tau(c_t)) \ldots

Hence, we have the following result:

The permutation \sigma^\prime\in S_n is a conjugate of \sigma \in S_n if and only if they have the same cycle structure.

The astute reader may point out that we’ve only demonstrated (LHS implies RHS). The converse is really quite easy to show, as the following example should amply illustrate:

Example : Show that the following permutations of S8 are conjugate, and find a \tau such that \sigma^\prime = \tau \sigma \tau^{-1}.

\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 3 & 7 & 6 & 2 & 5 & 1 & 8 & 4 \end{pmatrix},

\sigma^\prime = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ 5 & 1 & 8 & 3 & 2 & 4 & 7 & 6 \end{pmatrix}.

Answer : Write them as a product of disjoint cycles, thus giving (1,3,6)(2,7,8,4) and (1,5,2)(3,8,6,4) respectively. Thus, we can pick \tau \in S_8 which maps 1,3,6,2,7,8,4 to 1,5,2,3,8,6,4 respectively. This gives:

\tau = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 3 & 5 & 4 & 7 & 2 & 8 & 6 \end{pmatrix}.

Note that this is not the only answer!

Exercise : compute the number of solutions for \tau.

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One Response to Group Theory I.4 (Conjugate Permutations)

  1. vfp15 says:

    Thanks. Going through the exercises in Charles Pinter’s A Book of Abstract Algebra and this one had me stumped.

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