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lecture22 - 22 Quotient groups I 22.1 Definition of...

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22. Quotient groups I 22.1. Definition of quotient groups. Let G be a group and H a subgroup of G . Denote by G/H the set of distinct (left) cosets with respect to H . In other words, we list all the cosets of the form gH (with g G ) without repetitions and consider each coset as a SINGLE element of the newly formed set G/H . The set G/H (pronounced as G mod H ) is called the quotient set . Next we would like to define a binary operation * on G/H such that ( G/H, * ) is a group. It is natural to try to define the operation * by the formula gH * kH = gkH for all g, k G. (Q) Before checking group axioms, we need to find out whether * is at least well defined. Our first result shows that * is well defined whenever H is a normal subgroup. Theorem 22.1. Let G be a group and H a normal subgroup of G . Then the operation * given by (Q) is well defined. Proof. We need to show that if g 1 , g 2 , k 1 , k 2 G are such that g 1 H = g 2 H and k 1 H = k 2 H , then g 1 k 1 H = g 2 k 2 H . Recall that Theorem 19.2 (formulated slightly differently) asserts that given x, y G we have xH = yH ⇐⇒ x - 1 y H . Thus, we need to show the following implication if g - 1 1 g 2 H and k - 1 1 k 2 H, then ( g 1 k 1 ) - 1 g 2 k 2 H (!) So, assume that g - 1 1 g 2 H and k - 1 1 k 2 h . Then there exist h, h H such that g - 1 1 g 2 = h and k - 1 1 k 2 = h , and thus k 2 = k 1 h . Hence ( g 1 k 1 ) - 1 g 2 k 2 = k - 1 1 g - 1 1 g 2 k 2 = k - 1 1 hk 1 h = ( k - 1 1 hk 1 ) h . Since H is normal, k - 1 1 hk 1 H by Theorem 20.2, and so ( k - 1 1 hk 1 ) h H . Thus, we proved implication (!) and hence also Theorem 22.1. Remark: 1. The converse of Theorem 22.1 is also true, that is, if the operation * on G/H is well defined, then H must be normal. This fact is left as an exercise, but we will not use it in the sequel. 2. The book uses a different approach to defining the group operation in quotient groups (eventually, of course, the definition is the same, but initial justification is different). We could define the operation * by setting gH * kH to be the product of gH and kH as subsets of G (this operation was 1

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2 introduced in Lecture 19 and will be referred below as subset product ). With this definition, it is clear that the product is well defined, but it is not clear whether G/H is closed under it, that is, whether the subset product of two
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lecture22 - 22 Quotient groups I 22.1 Definition of...

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