By W. Edwin Clark

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COSETS AND LAGRANGE'S THEOREM 50 is a subgroup of U13 . Show that the subgroup H just de ned has exactly four di erent cosets in U13 . Note that if we list all the cosets 2H 22H 23H : : : 211 H 212 H it appears that there are 12 cosets. Show however that there are only four di erent cosets. Note that none of the cosets overlap, that is, if two cosets are di erent, then their intersection is the empty set. Also note that every element of U13 is in one and only one of the four di erent cosets and each coset of H has the same number of elements as H .

3 If and are disjoint cycles, then = . 25 Proof Let = (a1 ak ) and = (b1 b` ). Let fc1 cmg be the elements of n] that are in neither fa1 : : : ak g nor fb1 b`g. Thus n] = fa1 : : : ak g fb1 b` g fc1 cm g: We want to show (x) = (x) for all x 2 n]. To do this we consider rst the case x = ai for some i. Then ai 2= fb1 b`g so (ai ) = ai. Also (ai) = aj , where j = i + 1 or j = 1 if i = k. So also (aj ) = aj . Thus (ai ) = (ai ) = aj = (aj ) = ( (ai ) = (ai ): Thus, (ai ) = (ai ). It is left to the reader to show that (x) = (x) if x = bi or x = ci, which will complete the proof.

From the de nition, one may easily show that a subgroup H is a group in its own right with respect to this binary operation. Many examples of groups may be obtained in this way. In fact, in a way we will make precise later, every nite group may be thought of as a subgroup of one of the groups Sn. ; ; 31 CHAPTER 4. 2 Prove that if G is any group, then 1. feg G. 2. G G. The subgroups feg and G are said to be trivial subgroups of G. 3 (a) Determine which of the following subsets of S4 are subgroups of S4 .

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