72 1. ALGEBRAIC THEMES Two groups G and H are said to be isomorphic if there is a bijective map f W H ! G between them that makes the multiplication table of one group match up with the multiplication table of the other. The map f is called an isomorphism . (The requirement on f is this: Given a;b;c in H , we have c D ab if and only if f.c/ D f.a/f.b/ .) For another example, the permutation group S 3 is isomorphic to the group of symmetries of an equilateral triangular card, as is shown in Exer-cise 1.5.2 . Another simple way in which two groups may be related is that one may be contained in the other. For example, the set of symmetries of the square card that do not exchange top and bottom is f e;r;r 2 ;r 3 g ; this is a group in its own right. So the group of symmetries of the square card contains the group f e;r;r 2 ;r 3 g as a subgroup . Another example: The set of eight invertible 3-by-3 matrices f E;A;B;C;D;R;R 2 ;R 3 g introduced in Section 1.4 is a group under the operation of matrix multiplication; so it
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.