Prof. Alexandru Suciu
MATH 3175 Group Theory Fall 2010
Practice Quiz 1
1. Let d= gcd(20,24).
(a) Find d.
(b) Find a pair of integers sand tsuch that 20s+ 24t=d.
(c) Find the general solution for all the pairs of integers sand tsuch that 20s+24t=d.
2. Suppose a=p4
1p5
2p3and b=p3
2p9
3p4p8
5, where p1, . . . , p5are distinct primes.
(a) Find gcd(a, b).
(b) Find lcm(a, b).
(c) Check that gcd(a, b)·lcm(a, b) = ab.
3. Determine whether the following Latin square is the Cayley table of a group. If that’s
not the case, give a reason why not. If that’s the case, give an example of a group
whose Cayley table is this one.
e a b c d
e e a b c d
a a b d e c
b b d c a e
c c e a d b
d d c e b a
4. Determine whether the following Latin square is the Cayley table of a group. If that’s
not the case, give a reason why not. If that’s the case, give an example of a group
whose Cayley table is this one.
e a b c d
e e a b c d
a a c e d b
b b d c a e
c c e d b a
d d b a e c
