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Unformatted text preview: NOTES ON GROUP THEORY The goal of these notes is to introduce some of the basic concepts which come up in Math 113. As with the analysis/topology material, the emphasis will be placed on studying examples of algebra proofs and the specific techniques that they involve, rather than fully learning everything there is to know about groups. Here we have some basic definitions and propositions, which we will build on in class and in the homework. Definition 1. A nonempty subset H of a group G is said to be a subgroup of G if (i) For any x H , x- 1 H , and (ii) For any x,y H , xy H . Proposition 1. Let H be a subgroup of a group G . Then e H , where e is the identity element of G . In particular then, H is itself a group under the same operation as G . Proof. Let x be any element of H . Since H is a subgroup of G , x- 1 H . Thus e = xx- 1 is also in H . Proposition 2. Let G be a group and let H be a nonempty subset of G . Then H is a subgroup of G if and only if gh- 1 H for any g,h H . Proof. Suppose that H is a subgroup of G and let g,h H . Since H is a subgroup, we know that h- 1 H . Then, also since H is a subgroup, we know that gh- 1 H ....
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