Math 417: Abstract Algebra Homework 4
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Homework 4
Math 417: Abstract Algebra
(Exercises are taken from Algebra: Abstract and Concrete, Edition 2.6 by Frederick M. Goodman.)
1. Determine whether there is an isomorphism between Φ(5) and Φ(8).
2. Let (M, ·) be a monoid. Say that a subset N ⊆ M is a submonoid if (1) the product a,b →7 ab on M restricts to a function N × N → N such that (2) N with the restricted operation is a monoid.
a. Show that a subset N of a monoid M is a submonoid if and only if (i) if a,b ∈ N, then ab ∈ N, and (ii) there exists an element e′ ∈ N such that e′ a = a = ae′ for all a ∈ N.
b. Let M := Mat 2×2(R) be the monoid of 2 × 2 real matrices, with operation given by matrix multi-plication. Let N be the subset of M consisting of all matrices of the form （ 0(a) 0(0) ). Show that N is a submonoid of M.
c. Show that if N is a submonoid of M, then the identity element of N might not be the same as the identity element of M. (This is different from the case of subgroups, where we could show that the identity element of the subgroup must be equal to the identity element of the larger group.)
3. Exercise §2.2.4 Let H ⊆ S4 be the subset consisting of: all 3-cycles, all products of disjoint 2-cycles, and the identity.
a. Show that U := {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} is a subgroup of S4 .
b. Show that the product of any two 3 -cycles in S4 is either: the identity, a 3-cycle, or a product of two disjoint 2-cycles (see book for hints).
c. Show that the product of a 3-cycle in S4 with any product of two disjoint 2-cycles is a 3-cycle.
d. Show that H is a subgroup of S4 .
4. Exercise §2.2.10 Φ(14) is cyclic of order 6. Which elements of Φ(14) are generators? What is the order of each element of Φ(14)?
5. Let A = 
1 0(1) ) , B =（ 1(0) 
1(1) ). Show that as elements of the group GL2(R), the element A has order 4 and the element B has order 3 and that the element C := AB has infinite order.
6. Exercise §2.2.24 Suppose that a group G of order 20 has at least three elements of order 4. Can G be cyclic? What if G has exactly two elements of order 4?
7. Exercise §2.3.6 Find a subgroup of D6 that is isomorphic to D3 .
8. Exercise §2.3.7 Find a subgroup of D6 that is isomorphic to the symmetry group of the rectangle.
2023-08-18