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Section 14.1 Pre-read

Subsection Conjugate elements

Definition 14.1.1. Conjugate elements.

Group elements \(x\) and \(y\) are called conjugate if there exists a third group element so that

\begin{equation*} y = \inv{g} x g \text{.} \end{equation*}

Example 14.1.2. Conjugacy in \(\GL_n(\R)\).

In AUMAT 120, we call two square matrices \(A\) and \(B\) similar when there exists an invertible matrix \(P\) so that

\begin{equation*} B = \inv{P} A P \text{.} \end{equation*}

If \(A\) (and hence \(B\)) is invertible, then all three matrices are elements of the group \(\GL_n(\R)\text{,}\) and in this case similar matrices becomes the same concept as conjugate elements.

Subsection Cycle structure

In our exploration of conjugacy in the symmetric group \(S_n\text{,}\) we will need the following concept. Recall that every element of \(S_n\) can be expressed as a product of disjoint cycles.

Definition 14.1.3. Same cycle structure.

Two elements of \(S_n\) are said to have the same cycle structure if, when each is expressed as a product of disjoint cycles, they have

  • the same number of \(1\)-cycles (i.e. fixed numbers);

  • the same number of \(2\)-cycles appearing;

  • the same number of \(3\)-cycles appearing;

  • and so on.

Example 14.1.4.

The following two elements of \(S_{11}\) have the same cycle structure.

\begin{gather*} \fourcycle{1}{2}{3}{4} \threecycle{5}{6}{7} \twocycle{8}{9}, \\ \fourcycle{1}{5}{9}{11} \threecycle{2}{4}{6} \twocycle{7}{10}. \end{gather*}

Note: As usual, the one-cycles have been omitted. But since we are assuming these are elements of \(S_{11}\text{,}\) the one-cycles that should appear in the first element are \(\onecycle{10}\) and \(\onecycle{11}\text{,}\) and the one-cycles that should appear in the second element are \(\onecycle{3}\) and \(\onecycle{8}\text{,}\)

Subsection Centre of a group

Definition 14.1.5. Centre of a group.

Given a group \(G\text{,}\) the centre of \(G\) is defined to be the collection of all elements that commute with every other element, and we write \(\Zntr(G)\) for this collection. That is, \(\Zntr(G)\) consists of those elements \(x\) in \(G\) so that

\begin{equation*} g x = x g \end{equation*}

is true for every \(g\) in \(G\text{.}\)