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
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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