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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11.1.2 Groups Homomorphisms
Let GL(2,11) be the group of all invertible 2 × 2 matrices with entries in Z₁1, with group operation given my matrix multiplication. Consider the following two matrices in this group (where an entry listed as k is shorthand for [k]11): 3 3 10 4- (1₂0). B- (B) A = B= 8 8 (ii) Consider the subset of GL(2,11) given by G= {Am B : m, n = Z}. Show that G is a subgroup of GL(2, 11).
Suppose that T: R" → Rm is defined by T() = A for each of the following matrices below: [40] E = 0 0 02 0 D = 0 -1 0 0 0 0.5 [201] F = 10 3 01 (a) Rewrite T: R" → Rm with correct numbers for m and n for each transformation. What is the domain and codomain of each transformation? (b) Find some way to explain in words and/or graphically what each transformation does as it takes vectors from R to Rm. You might find it t helpful to try out a few input vectors and see what their image is under the transformation. This might be difficult, but an honest effort will give you credit. (c) For the transformation, can you get any output vector? (Any vector in R™) i. If so, explain why you can get any vector in Rm. ii. If not, give an example of an output vector you can't get with the transformation and explain why.
(4) Chapter 2, exercise 6.9: Prove that a group and its opposite group Gº (Exercise 2.6) are isomorphic.
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Suppose that B = S-¹AS for some n x n matrices A, B and some n x n invertible matrix S. a. Show that if is in ker B, then Sa is in ker A. b. Show that the linear transformation T (*) = S from ker B to ker A is an isomorphism. c. Show that nullity A = nullity B and rank A = rank B.
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