Let f=( 1 2 3 4 5 6 6 4 2 1 5 3 ) and g=(

step 1 We need to evaluate the following. So we're first going to evaluate G of 3. So we have G of 3 is

Let f=( 1 2 3 4 5 6 6 4 ...

Let $$ f=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 4 & 2 & 1 & 5 & 3 \end{array}\right) \text { and } g=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 5 & 6 & 2 & 4 & 1 \end{array}\right) \text { . } $$ Determine (a) $f \circ g$ and $g \circ f$ (b) $f^{-1}$ and $g^{-1}$ (c) $f^{2}, f^{5}$ (d) $f \circ g \circ f$ (e) $g^{3}$ and $f \circ g^{3} \circ f^{-1}$

Let
$$
f=\left(\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
6 & 4 & 2 & 1 & 5 & 3
\end{array}\right) \text { and } g=\left(\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
3 & 5 & 6 & 2 & 4 & 1
\end{array}\right) \text { . }
$$
Determine
(a) $f \circ g$ and $g \circ f$
(b) $f^{-1}$ and $g^{-1}$
(c) $f^{2}, f^{5}$
(d) $f \circ g \circ f$
(e) $g^{3}$ and $f \circ g^{3} \circ f^{-1}$

Key Concepts

Powers of a Permutation

Raising a permutation to a power involves composing the permutation with itself repeatedly. The exponentiation reveals structural properties such as the order of the permutation, which is the smallest positive power that results in the identity permutation. This is a key concept in understanding the cyclic subgroup generated by the permutation.

Conjugation in Permutation Groups

Conjugation involves transforming one permutation by another through the operation f ? g ? f?¹. This process is important because it preserves the cycle structure and order of elements within the permutation group. Conjugate permutations are considered structurally similar, and this concept is central to the study of symmetry and class equations in group theory.

Permutation Composition

Composition of permutations is the operation of performing one permutation after another. This operation is associative, and it defines the group structure on the set of all permutations of a given set. The order in which permutations are composed is crucial, because in general, permutation composition is not commutative.

Cycle Notation

Cycle notation is a compact way to represent a permutation by writing its elements as cycles. Instead of listing the full mapping of each element, it groups the elements that are permuted cyclically, allowing one to see the structure and order of the permutation more clearly.

Inverse of a Permutation

The inverse of a permutation is the permutation that 'undoes' the effect of the original permutation, mapping each element back to its original position. In cycle notation, the inverse of a cycle is formed by writing the cycle in reverse order, and for disjoint cycles, the inverse is computed by taking the inverse of each cycle.

Permutation

A permutation is a bijective function from a set onto itself. It rearranges the elements of the set in a one-to-one manner. In algebra, permutations are used to study the symmetry and structure of finite sets, and they form the basis of permutation groups, which are central objects in group theory.

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