The Cayley Representation Theorem.

Question

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.

Answer 1

You ask for a simple example, so here is one: Consider the group $\mathbb{Z} / n\mathbb{Z}$. This is a finite abelien group under addition of order $n$.

Let $$ \phi: \mathbb{Z} / n\mathbb{Z} \longrightarrow S_n $$ given by $$ \phi([1]) = (1\; 2\; 3\; \dots \; n ). $$ So $$ \phi([m]) = (1\; 2\; 3\; \dots \; n )^m. $$ Then $\phi$ is a injective. Note that $S_n$ is a multiplicative group.

Note that all you had to do here was to find an element of order $n$ in $S_n$. In fact, the above example is easily extended to embedding any cyclic group in some permutation group (since any cyclic group is isomorphic to a $\mathbb{Z} / n\mathbb{Z}$ for some $n$).

Answer 2

For example, say you have a finite Abelian group which is a product of cyclic groups of orders $q_1,q_2,\ldots,q_n$. Then let $N = \sum_j q_j$ and consider the permutation group $S_N$. For each $q_j$, choose a subset of $\{1,2,\ldots,N\}$ of size $q_j$ where all the subsets are disjoint. Then consider the subgroup generated by a set of cycles, where you have one cycle over each disjoint subset of characters that you picked. The subgroup generated by these cycles will be isomorphic to your Abelian group.