Question

9. Do only two (2) of the following: (a) Let G = GL(2, R) and H = {A ? G| det(A) = 7^k, k ? Z}. Prove that the subset H is a normal subgroup of G. (Note that you need to justify H is a subgroup of G.) (b) Let a and b be nonidentity elements of different orders in a group G of order 21. Prove that the only subgroup of G that contains a and b is G itself. (c) Let G be a finite non-abelian group. Show that the factor group G/Z(G) is not cyclic (You should show all details). (d) Let H be a subgroup of D15 such that the order of H is 5. Prove that H must be a cyclic subgroup of D15.

          9. Do only two (2) of the following:
(a) Let G = GL(2, R) and H = {A ? G| det(A) = 7^k, k ? Z}. Prove that the subset H is
a normal subgroup of G. (Note that you need to justify H is a subgroup of G.)
(b) Let a and b be nonidentity elements of different orders in a group G of order 21. Prove
that the only subgroup of G that contains a and b is G itself.
(c) Let G be a finite non-abelian group. Show that the factor group G/Z(G) is not cyclic
(You should show all details).
(d) Let H be a subgroup of D15 such that the order of H is 5. Prove that H must be a
cyclic subgroup of D15.

Show more…

9. Do only two (2) of the following:
(a) Let G = GL(2, R) and H = A ? G| det(A) = 7^k, k ? Z. Prove that the subset H is
a normal subgroup of G. (Note that you need to justify H is a subgroup of G.)
(b) Let a and b be nonidentity elements of different orders in a group G of order 21. Prove
that the only subgroup of G that contains a and b is G itself.
(c) Let G be a finite non-abelian group. Show that the factor group G/Z(G) is not cyclic
(You should show all details).
(d) Let H be a subgroup of D15 such that the order of H is 5. Prove that H must be a
cyclic subgroup of D15.

Added by Rodney W.

A Book of Abstract Algebra

Charles C. Pinter 1982 Edition

Chapter 5

Instant Answer

Solved by Expert Vincenzo Zaccaro

10/27/2023

Step 1

First, we need to show that H is a subgroup of G. To do this, we need to show that H is closed under the group operation and that it contains the identity element and inverses. Let A, B ∈ H. Then det(A) = det(B) = 7. We know that det(AB) = det(A)det(B) = 7 * 7 = Show more…

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Do only two of the following: Let G = GL(2,R) and H = {A ∈ G | det(A) = 7^k, k ∈ Z}. Prove that the subset H is a normal subgroup of G. (Note that you need to justify H is a subgroup of G.) Let a and b be nonidentity elements of different orders in group G of order 21. Prove that the only subgroup of G that contains a and b is G itself. Let G be a finite non-abelian group. Show that the factor group G/Z(G) is not cyclic. (You should show all details.) Let H be a subgroup of D15 such that the order of H is 5. Prove that H must be a cyclic subgroup of D15.

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