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We can compose two symmetries and the result is another symmetry. The reason for this is that the composition of two one-to-one mapping is again a one-to-one mapping. Thus we can consider this composition of mappings as a binary operation that combines two symmetries into one. Theorem 10. The set of symmetries of a regular polygon, with binary operation defined as the composition of symmetries is a group. Proof. All that we need to do is check that this set and binary operation meet the definition of a group.

Example: From linear algebra, matrix A is row equivalent to matrix B if B is obtained from A by a finite number of elementary row operations. We will show that row equivalence is an equivalence relation. Row equivalence is reflexive. A is row equivalent to itself as we can start with A, perform no row operations, and end with A. Row equivalence is symmetric. If A is row equivalent to B then we obtain B from A by a finite number of elementary row operations. Each of these row operations can be reversed by an elementary row operation.

Elements. ☐ It is important to note how quickly the size of Sn increases as n increases. For instance the order of S5 is 120, and the order of S10 is 3, 628, 800 . The factorial goes a long way. Due to the order of Sn for relatively small values of n , we will look at some symmetric groups of the lowest orders. 1 Examples of Sn Example: For n = 1 there is one permutation from the set A = {1} to A. This is the identity permutation, and so S1 consists of {ι} . It is clear that ι ◦ ι = ι . com 56 An Introduction to Abstract Algebra Group Theory Example: For n = 2 there are two permutations from the set A = {1, 2} to A.

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