This file includes some HTML versions of previous finals exams I have given in Math 420 (Abstract Algebra I). Note that it uses HTML tags for subscripts and superscripts, so it is difficult to decipher with an earlier viewer.

(b) Compute gcd(1776,1492).

(c) Show that if a,b,c are positive integers, then gcd(a,bc) = 1 if and only if gcd(a,b) = 1 and gcd(a,c) = 1.

**2.**
(a) Find phi(32) (the Euler phi-function evaluated at 32).

(b) Use the Euclidean algorithm to find
[5]_{32}^{-1} in Z_{32}^{x}.

(c) Find all powers of
[5]_{32} in Z_{32}^{x}.

**3.**
Define the following permutations (in two-row notation).

sigma = ( 1 2 3 4 5 6 7 4 6 1 3 2 5 7 ) tau = ( 1 2 3 4 5 6 7 3 2 4 6 7 1 5 )(a) Compute the products sigma tau and tau sigma.

(b) Write each of sigma, tau, sigma tau, and tau sigma as a product of disjoint cycles.

(c) Compute the order (in S

**4.**
Let S be the set of all ordered pairs (m,n) of positive integers m, n.
On S, define

(m_{1},n_{1}) ~ (m_{2},n_{2}) if
m_{1} + n_{2} = m_{2} + n_{1}.

(a) Show that ~ defines an equivalence relation on S.

(b) On the equivalence classes S/~, define an addition as follows:

[(m_{1},n_{1})] + [(m_{2},n_{2})] =
[(m_{1}+m_{2},n_{1}+n_{2})].

Show that there is an identity element for this addition.
Then find a formula for the additive inverse of [(m,n)].
(You may assume that the formula for addition
gives a well-defined and associative binary operation.)

**5.**
(a) State these definitions: group; subgroup.

(b) State Lagrange's theorem.

(c) Let G be a group, and let H be a nonempty subset of G.
Suppose that if x and y are any elements of H,
then xy^{-1} is in H.
Show that H must be a subgroup of G.

**6.**
Let m and n be positive integers with gcd(m,n) = 1.

Define f:Z_{mn} -> Z_{m} x Z_{n} by

f([x]_{mn}) = ([x]_{m},[x]_{n}),
for all [x]_{mn} in Z_{mn}.

(a) Show that f is a well-defined function.

(b) State the definition of an isomorphism of groups.

(c) Show that f is an isomorphism.

**7.**
For each of the following,
either indicate that the statement is true,
or give a counterexample if the statement is false.

(a) If G is a finite group of order n,
then every element x of G satisfies the equation x^{n} = e.

(b) If G is a finite group of order n,
then every element (except the identity e) has order n.

(c) If G is a finite group of order n,
then there is at least one element of G that has order n.

(d) If G is a finite group of order n, and n is prime,
then there is at least one element of G that has order n.

(e) If a and b are group elements of order m and n, respectively,
then the element ab has order lcm[m,n].

**8.**
Prove ONE of the following theorems from the text.

I. Every subgroup of a cyclic group is cyclic.

II. If G is a cyclic group of order n,
then G is isomorphic to Z_{n}.

III. Every group is isomorphic to a group of permutations.

2x is congruent to 9 (mod 15)

x is congruent to 8 (mod 11)

**2.**
Find ([91]_{501})^{-1}
(in the multiplicative group Z_{501}^{x}).

**3.**
Let sigma = (2,4,9,7)(6,4,2,5,9)(1,6)(3,8,6) in S_{9}.

(i) Write sigma as a product of disjoint cycles.

(ii) What is the order of sigma?

(iii) Compute the inverse of sigma.

**4.**
Let G be a group.

(a) State the definition of a subgroup of G.

(b) State a result that tells you which conditions to check
when determining whether or not a subset of G is a subgroup of G.
Use this result in proving part (c).

(c) Let H and K be subgroups of G.
Prove that the intersection of H and K, defined as

{ g in G | g is in H and g is in K }

is a subgroup of G.

**5.**
(a) State the definition of a cyclic group.

(b) Write out ONE of the following proofs from the text:

I. Any subgroup of a cyclic group is cyclic.

II. If G is a cyclic group of order n,
then G is isomorphic to Z_{n}.

**6.**
Do ONE of the following problems.

I. Find all subgroups of Z_{11}^{x},
and give the lattice diagram which shows the
inclusions between them.

II. Show that the three groups
Z_{6},
Z_{9}^{x}, and
Z_{18}^{x} are isomorphic to each other.

**7.**
Let G be the subgroup of GL_{3}(R) consisting of all matrices
of the form

_ _ | | | 1 a b | | | such that a,b are real numbers. | 0 1 0 | | | | 0 0 1 | |_ _|Show that G is a subgroup of GL

(Remember that the general linear group consists of all invertible matrices, and the operation is matrix muliplication.)

**8.**
Show that the group G in problem 7
is isomorphic to the direct product R x R.

(Remember that R denotes the group of all real numbers,
and the operation is ordinary addition.)

**2.**
(20 pts)
(a) Is 7^{123} + 1 divisible by 3?

(b) What is the last digit in the decimal expansion of 4^{123}?

**3.**
(20 pts)
Define the following permutations (in two-row format).

sigma = ( 1 2 3 4 5 6 7 and tau = ( 1 2 3 4 5 6 7 5 6 3 1 4 7 2 ) 7 1 2 6 5 4 3 ).(a) Write sigma, tau, sigma tau, and tau sigma as products of disjoint cycles.

(b) Find the order of each of sigma, tau, sigma tau, and tau sigma.

**4.**
(20 pts)
Define f: Z_{n} -> Z_{m} by
f([x]_{n}) = [kx]_{m}.
Show that the formula f defines a function
if and only if m | kn.
Find conditions on n, m, k that determine
when f is a one-to-one correspondence.

**5.**
(20 pts)
Let G be a group and let H be a subgroup of G .
For elements x,y in G,
define x ~ y if y^{-1}x is in H.
Check that ~ defines an equivalence relation on G.

**6.**
(25 pts)
(a) Define the following terms: group; cyclic group;
order of an element of a group.

(b) State the following theorems:
Lagrange's theorem (about the order of a subgroup);
Cayley's theorem (about groups of permutations).

**7.**
(25 pts)
Let G be any cyclic group.
Prove that G is isomorphic to either Z or Z_{n},
for some positive integer n.

**8.**
(20 pts)
(a) Show that Z_{5}^{x} is isomorphic to
Z_{10}^{x}.

(b) Show that Z_{30}^{x} is *not* isomorphic to
Z_{24}^{x}.

**9.**
(30 pts)
Let G and G' be groups,
and let f : G -> G' be a function
(not required to be either one-to-one or onto)
such that f(ab) = f(a)f(b) for all a,b in G.

(a) Let e and e' denote the identity elements of G and G', respectively.
Show that f(e) = e',
and that f(x^{-1}) = (f(x))^{-1} for all x in G.

(b) Show that the subset { x in G | f(x)=e' } is a subgroup of G.

(c) Show that the subset
{ y in G' | y = f(x) for some x in G } is a subgroup of G'.

**1.**
Let a and b be nonzero integers.
Prove that gcd(a,b) = 1 if and only if
gcd(a+b,ab) = 1.

**2.**
Solve the following system of congruences:

2x is congruent to 7 (mod 15)

3x is congruent to 5 (mod 14)

**3.**
Let G be any cyclic group.
Prove that G is isomorphic to either Z or Z_{n},
for some positive integer n.

**4.**
Let G be a group and let H be a subgroup of G.
For any element a in G, define

Ha = { x in G | x = ha for some h in H }.

Prove that the collection of all such subsets partitions G.

**5.**
Let sigma = (2, 4, 9, 7,) (6, 4, 2, 5, 9) (1, 6) (3, 8, 6)
in the symmetric group S_{9}.

(i) Write sigma as a product of disjoint cycles.

(ii) What is the order of sigma?

(iii) Compute the inverse of sigma.

**6.**
Let G be a group.
Show that G is abelian if and only if
(ab)^{-1} = a^{-1}b^{-1}
for all a,b in G.

**7.**
If a nontrivial group G has no proper nontrivial subgroups,
prove that G is cyclic and that the order of G is a prime number.

**8.**
Let G be any group.
In the proof of Cayley's theorem, for each a in G a function

lambda_{a} : G -> G is defined by lambda_{a}(x) = ax,
for all x in G.

(a) Prove that each of these functions defines a permutation of G.

(b) Prove that the subset
{ lambda_{a} | a in G }
is a subgroup of Sym(G).

**9.**
Let G be a group and let H and K be subgroups of G.
Prove that the intersesction of H and K is a subgroup of G.

**10.**
Define the following terms: one-to-one function; onto function;
group; cyclic group.

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