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?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 41 belongs to the center of
(GL(2, R), ·) because (4I)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?

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