College Algebra Exam Review 114

College Algebra Exam Review 114 - 124 2 BASIC THEORY OF...

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Unformatted text preview: 124 2. BASIC THEORY OF GROUPS If the groups are infinite, we have to use another approach, which is discussed in the Exercises. I Definition 2.5.11. For any group G , the center Z.G/ of G is the set of elements that commute with all elements of G , Z.G/ D fa 2 G W ag D ga for all g 2 G g: You are asked in the Exercises to show that the center of a group is a normal subgroup, and to compute the center of several particular groups. Exercises 2.5 2.5.1. Check that the left cosets of the subgroup K D fe; .123/; .132/g in S3 are eK D .123/K D .132/K D K .12/K D .13/K D .23/K D f.12/; .13/; .23/g and that each occurs three times in the list .gK/g 2S3 . Note that K is the subgroup of even permutations and the other coset of K is the set of odd permutations. 2.5.2. Suppose K Â H Â G are subgroups. Suppose h1 K; : : : ; hR K is a list of the distinct cosets of K in H , and g1 H; : : : ; gS H is a list of the distinct cosets of H in G . Show that fgs hr H W 1 Ä s Ä S; 1 Ä r Ä Rg is the set of distinct cosets of H in G . Hint: There are two things to show. First, you have to show that if .r; s/ ¤ .r 0 ; s 0 /, then gs hr K ¤ gs 0 hr 0 K . Second, you have to show that if g 2 G , then for some .r; s/, gK D gs hr K . 2.5.3. Try to extend the idea of the previous exercise to the case where at least one of the pairs K Â H and H Â G has infinite index. 2.5.4. Consider the group S3 . (a) (b) Find all the left cosets and all the right cosets of the subgroup H D fe; .12/g of S3 , and observe that not every left coset is also a right coset. Find all the left cosets and all the right cosets of the subgroup K D fe; .123/; .132/g of S3 , and observe that every left coset is also a right coset. ...
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