12. (a) Is the subset { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a subgroup of...

12. (a) Is the subset { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a subgroup of the direct product group ( V x Z18 )? (V is the Klein four group.) Carefully explain or justify your answer.

(b) Is the subgroup { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a normal subgroup of the direct product group ( V x Z18 )? Carefully explain or justify your answer.

Solutions

Expert Solution

upvote

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

(a) Suppose K is a subgroup of H, and H is a subgroup of G. If...

(a) Suppose K is a subgroup of H, and H is a subgroup of G. If |K|= 20 and |G| = 600, what are the possible values for |H|? (b) Determine the number of elements of order 15 in Z30 Z24.

1. Let N be a normal subgroup of G and let H be any subgroup of...

1. Let N be a normal subgroup of G and let H be any subgroup of G. Let HN = {hn|h ∈ H,n ∈ N}. Show that HN is a subgroup of G, and is the smallest subgroup containing both N and H.

Consider G = (Z12, +). Let H = {0, 3, 6, 9}. a. Show that H...

Consider G = (Z12, +). Let H = {0, 3, 6, 9}. a. Show that H is a subgroup of G. b. Find all the cosets of H in G and denote this set by G/H. [Note: If x ∈ G then H +12 [x]12 = {[h + x]12?? | [h]12 ∈ H} is the coset generated by x.] c. For H +12 [x]12, H +12 [y]12 ∈ G/H define (H+12[x]12)⊕(H+12[y]12) by(H+12 [x]12)⊕(H+12 [y]12)=H+12 [x+y]12. d. Show that ⊕ is...

Let H be the subset of all skew-symmetric matrices in M3x3 a.) prove that H is...

Let H be the subset of all skew-symmetric matrices in M3x3 a.) prove that H is a subspace of M3x3 by checking all three conditions in the definition of subspace. b.) Find a basis for H. Prove that your basis is actually a basis for H by showing it is both linearly independent and spans H. c.) what is the dim(H)

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

show that if H is a p sylow subgroup of a finite group G then for...

show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G

(1) Recall on February 6 in class we discussed e 0 + e 2πi/n + e...

(1) Recall on February 6 in class we discussed e 0 + e 2πi/n + e 4πi/n + · · · + e 2(n−1)πi/n = 0 and in order to explain why it was true we needed to show that the sum of the real parts equals 0 and the sum of the imaginary parts is equal to 0. (a) In class I showed the following identity for n even using the fact that sin(2π − x) = − sin(x):...

Question