S TUDY S ESSION 4
M ATH 120A FALL 2016
1. a. Let G be a group and let g be an element of G. Define, in your own words, the order of g in
G.
b. Find the order of each element of (Z12 , +12 ).
c. Find the order of each 12th root of unity in the group (C
{0}, ·).
d. If G = hai is a cyclic group of order 12, list all the elements in G and find the order of each
element.
2. Let G be a group and let g be an element of G. Prove that g and g
1
have the same order.
3. Let GL2 (R) be the group of 2 ⇥ 2 invertible matrices, with multiplication. (The elements of
GL2 (R) have real entries and non-zero determinant.) Consider the matrix:
✓
◆
1 1
A=
.
0 1
a. Find the cyclic subgroup H of GL2 (R) generated by the matrix A:
H = hAi = {Ak : k 2 Z}.
b. Find a familiar group isomorphic to H. Explicitly provide an isomorphism (and check that
the given map is, indeed, an isomorphism).
c. Can you find a second isomorphism from H to this familiar group?
4. In this problem, we always consider Zn to have as group operation addition modulo n: +n . For
all elements (g, h) in Z2 ⇥ Z2 , find the order of g in Z2 , the order of h in Z2 and the order of
(g, h) in Z2 ⇥ Z2 .
(g, h)
([0]2 , [0]2 )
([0]2 , [1]2 )
([1]2 , [0]2 )
([1]2 , [1]2 )
order of g in G order of h in H order of (g, h) in G ⇥ H
Repeat with Z2 ⇥ Z3 .
(g, h)
order of g in G order of h in H order of (g, h) in G ⇥ H
([0]2 , [0]3 )
([0]2 , [1]3 )
([0]2 , [2]3 )
([1]2 , [0]3 )
([1]2 , [1]3 )
([1]2 , [2]3 )
In general, for all groups G and H, make a conjecture about the order of the element (g, h) in
G ⇥ H, compared to the orders of g in G and h in H.
5. If G0 is a subgroup of G and H0 is a subgroup of H, prove that G0 ⇥ H0 is a subgroup of G ⇥ H.
Is every subgroup of this form?
6. Let G be a group and let g be an element of G of finite order. Let n
1 be an integer. Show that
g n = e () the order of g divides n.
(Hint: To prove =), divide n by the order of g and show that the remainder must be zero. Use
the contrapositive and properties of remainders.)
7. Prove that if a group G has no non-trivial proper subgroup, then it is cyclic. (Hint. Use the
contrapositive.)
8. Let G be a group, and let a be an element of G of order 24. List all possible subgroups of hai.
(For each subgroup, give a generator and specify the order.)