Math 307 Abstract Algebra Homework 3
Sample Solution
1. Let G be a group and a G.
(1) Show that C (a) = cfw_g G : ag = ga is a subgroup of G.
(2) Show that Z (G) = aG C (a).
Solution. (1) Let a G. Clearly, e C (a) because ae = ea. If x, y C (a), then (xy
Math 307 Abstract Algebra
Homework 4
Due: 26 Sept., 2013
Solve the following problems. Please use complete sentences and good grammar.
Each problem is 4 points.
1. Consider = (13256)(23)(46512).
(a) Express as a product of disjoint cycles.
(b) Express as
Math 307 Abstract Algebra
Homework 7
Due: Noon, Friday, 25 Oct., 2013
Four points for each question.
1. (a) Let G = cfw_3a 6b 10c : a, b, c Z under multiplication. Show that G is isomorphic to
3 6 10 .
(b) Let H = cfw_3a 6b 12c : a, b, c Z under multiplic
Math 307 Abstract Algebra
Homework 9
Due: Noon, Nov.8, 2013
Four points for each question.
1. (a) Give an example of a subset of a ring that is a subgroup under addition but not a subring.
(b) Give an example of a nite non-commutative ring.
2. Show that i
Math 307 Abstract Algebra
Homework 10
Due: Noon, Friday, 15 Nov., 2013
Four points for each question. Explain your examples if the questions ask for examples.
1. (a) Give an example to show that the characteristic of a subring of a ring R may be dierent
f
Math 307 Abstract Algebra Homework 3
your name
Due: 19 Sept., 2013
Solve the following problems. Please use complete sentences and good grammar.
Each problem is 4 points.
1. Let G be a group and a G.
(1) Show that C(a) = cfw_g G : ag = ga is a subgroup of
Math 307 Abstract Algebra
Homework 6
Due: Noon, Friday, 11 Oct., 2013
Four points for each question.
1. If r is a divisor of m and s is a divisor of n, nd a subgroup of Zm Zn that is isomorphic
to Zr Zs .
2. (a) Prove that R R under addition in each compo
Math 307 Abstract Algebra
Homework 2
Due: Sept. 12, 2013
Solve the following problems (4 points each).
1. Prove that the set of all 2 2 matrices with entries from R and determinant 1 is a group
under matrix multiplication.
2. Prove that the set of all rat
Math 307 Abstract Algebra
Homework 4
Due: Noon, Friday, 4 Oct., 2013
Four points for each question.
1. (a) Let H = (1, 2) S3 . Write down all the left cosets of H in S3 , and also the right cosets
of H in S3 .
(b) Let nZ = cfw_nk : k Z Z under addition. D
Math 307 Abstract Algebra
Homework 1
Due: Sept. 5, 11:00 a.m.
Solve the following problems. Use complete sentences and good grammar. Each problem is 4 points.
1. Suppose that a|c and b|c. If a and b are relatively prime, show that ab|c. Show, by example,
Math 307 Abstract Algebra
Homework 2
Sample Solution
1. Prove that the set of all 2 2 matrices with entries from R and determinant 1 is a group
under matrix multiplication.
Let M be the set of all 2 2 matrices with real entries such that for A M , det(A)
Math 307 Abstract Algebra
Homework 1
Solution 1
1. Suppose that a|c and b|c. If a and b are relatively prime, show that ab|c. Show, by example, that if
a, b are not relatively prime, then ab need not divide c.
Since a|c and b|c, there exist m, n Z such th
Math 307 Abstract Algebra
Homework 12
Due: Noon, December 6, 2013
1. (a) Write x3 + 6 Z7 [x] as a product of irreducible polynomials over Z7 .
(b) Write x3 + x2 + x + 1 Z2 [x] as a product of irreducible polynomials over Z2 .
2. (a) Find all units in Z4 [
Math 307 Abstract Algebra
Homework 5
Sample solution
1. (a) Let H = (1, 2) S3 . Write down all the left cosets of H in S3 , and also the right cosets
of H in S3 .
Solution (1, 3)H = (1, 2, 3)H = cfw_(1, 3), (1, 2, 3), (2, 3)H = (1, 3, 2)H = cfw_(2, 3), (1
Math 307 Abstract Algebra
Homework 6
Sample solution
1. If r is a divisor of m and s is a divisor of n, nd a subgroup of Zm Zn that is isomorphic
to Zr Zs .
Solution. Let a = m/r, b = n/s, H = cfw_(pa, qb) : p, q Z, and : Zr Zs H dened by
(p, q ) = (pa, q
Math 307 Abstract Algebra
Homework 7
Sample solution
1. (a) Let G = cfw_3a 6b 10c : a, b, c Z under multiplication. Show that G is isomorphic to
3 6 10 .
(b) Let H = cfw_3a 6b 12c : a, b, c Z under multiplication. Show that G is NOT isomorphic to
3 6 12 .
Math 307 Abstract Algebra
Homework 8
Sample solution
1. (a) Let G be the group of nonzero real numbers under multiplication. Suppose r is a positive
integer. Show that x xr is a homomoprhism. Determine the kernel, and determine r so
that the map is an iso
Math 307 Abstract Algebra
Homework 8
Sample solution
1. (a) Let G be the group of nonzero real numbers under multiplication. Suppose r is a positive
integer. Show that x xr is a homomoprhism. Determine the kernel, and determine r so
that the map is an iso
Math 307 Abstract Algebra
Homework 9
Sample solution
1. (a) Give an example of a subset of a ring that is a subgroup under addition but not a subring.
(b) Give an example of a nite non-commutative ring.
Solution. (a) Let H = (2, 3) Z Z. Then H = cfw_(2k,
Math 307 Abstract Algebra
Homework 10
Sample solution
1. (a) Give an example to show that the characteristic of a subring of a ring R may be dierent
from that of R.
(b) Show that the characteristic of a subdomain of an integral domain D is the same as tha
Math 307 Abstract Algebra
Homework 11
Due: Noon, Tuesday 26 Nov., 2013
Four points for each question.
1. List all the polynomials of degree 2 in Z2 [x]. Which of these are equal as functions from Z2
to Z2 , i.e., p(x) = q(x) for x = 0, 1?
2. If : R S is a
Math 307 Abstract Algebra
Homework 11
Sample Solution
1. List all the polynomials of degree 2 in Z2 [x]. Which of these are equal as functions from Z2
to Z2 , i.e., p(x) = q (x) for x = 0, 1?
Solution. f1 (x) = x2 , f2 (x) = x2 + x, f3 (x) = x2 + 1, f4 (x
Math 307 Abstract Algebra
Homework 12
Sample solution
1. (a) Write x3 + 6 Z7 [x] as a product of irreducible polynomials over Z7 .
(b) Write x3 + x2 + x + 1 Z2 [x] as a product of irreducible polynomials over Z2 .
Solution. (a) Let f (x) = x3 + 6, then f