Name:
Problem Set 6
Math 415 Honors, Fall 2014
Due: Tuesday, October 28.
Review Sections 14, 15 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Tuesday, October 28. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Let G = (Z20 × Z30 )/h(4, 3)i. This is a finite abelian group, so it is isomorphic to a product of
cyclic groups of prime-power order. Find this product expression. Prove your answer.
2. Same as in the previous exercise, for G = (Z10 × Z15 )/h(1, 1)i.
3. Let G = (Z × Z)/h(1, 1)i. This is a finitely generated abelian group, so it is isomorphic to a
direct product of cyclic groups of prime-power order, times some copies of Z. Find this product
expression, and provide the explicit isomorphism. (Do not forget to verify you actually have
an isomorphism).
4. Same as the previous exercise for G = (Z × Z)/h(5, 5)i.
5.
a. Give an example of a group G such that the identity is its only element of finite order,
with a normal subgroup H E G, such that all the elements of G/H have finite order.
b. Give an example of a group G and two normal subgroups H, K E G such that H K but
G/H is not isomorphic to G/K.
6. Show that R/Z U, where R and Z are the additive groups of real and integer numbers
respectively, and U is the multiplicative group of complex numbers of absolute value 1.
7. Let G be a group and H E G, with (G : H) = n. Prove that xn ∈ H for all x ∈ G.
8. Let G be a group and H, K normal subgroups. Show that H ∩ K E G. Do not forget to justify
that H ∩ K is a subgroup of G.
9. Let G be a finite group. Let H ≤ G, and suppose that H has the property that any subgroup K
of G that is different from H must have |H| , |K|. Show that H E G.
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10. Prove Theorem 15.16 from the book.
11.
a. Let G be a group, and Z(G) its center. Show that if G/Z(G) is cyclic, then G is abelian.
b. Let G be a nonabelian group of order pq, where p and q are prime. Show that G has
trivial center.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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