UA-AUG-AUMAT229 Pre-read

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

y = g^{- 1} x g .

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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

B = P^{- 1} A P .

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

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Subsection Cycle structure

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In our exploration of conjugacy in the symmetric group

S_{n},

we will need the following concept. Recall that every element of

S_{n}

can be expressed as a product of disjoint cycles.

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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.

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Example 14.1.4.

The following two elements of $S_{11}$ have the same cycle structure.

\begin{matrix} (1 2 3 4) (5 6 7) (8 9), \\ (1 5 9 11) (2 4 6) (7 10) . \end{matrix}

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

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Subsection Centre of a group

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Definition 14.1.5. Centre of a group.

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

g x = x g

is true for every $g$ in $G .$