8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set Z(G) consists of all elements of G that commute with every possible element of the group. For example, one can say that the matrix 4I belongs to the center of (GL(2, R), ·) because (41)A = A(4I) for all A in GL(2,R), since both sides are equal to 4A. (a) Show that, for every group G, the center Z(G) is a subgroup of G. (b) Find the center of (Z4,+) and (this part is moved to next homework) the center of D6, the dihedral group. (You should be able to tell from the group table.) (c) One could say that "the center Z(G) measures the abelian-ness of a group G". Please interpret this statement. 'Recall that a group (G, *) is called abelian if the operation * is commutative. Hint: What is Z(G) equal to when G is abelian?

8. Let (G, ) be a group. Define the center of G by Z(G) := {x € G: x a = a * x, Va E G}. The set Z(G) consists of all elements of G that commute with every possible element of the group. For example, one can say that the matrix 4I belongs to the center of (GL(2, R), ·) because (41)A = A(4I) for all A in GL(2,R), since both sides are equal to 4A. (a) Show that, for every group G, the center Z(G) is a subgroup of G. (b) Find the center of (Z4,+) and (this part is moved to next homework) the center of D6, the dihedral group. (You should be able to tell from the group table.) (c) One could say that "the center Z(G) measures the abelian-ness of a group G". Please interpret this statement. 'Recall that a group (G, ) is called abelian if the operation is commutative. Hint: What is Z(G) equal to when G is abelian?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 5E: The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of...

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