abstract algebra: a study guide for beginners - Northern Illinois ...

32 CHAPTER 3. GROUPS
64.† (a) Find the cyclic subgroup of S7 generated by the element (1, 2, 3)(5, 7).
(b) Find a subgroup of S7 that contains 12 elements. You do not have to list all of
the elements if you can explain why there must be 12, and why they must form a
subgroup.
65. (a) What are the possibilities for the order of an element of Z × 27 ? Explain your answer.
(b) Show that Z × 27
is a cyclic group.
3.3 Constructing Examples
The most important result in this section is Proposition 3.3.7, which shows that the set of
all invertible n × n matrices forms a group, in which we can allow the entries in the matrix
to come from any field. This includes matrices with entries in the field Zp, for any prime
number p, and this allows us to construct very interesting finite groups as subgroups of
GLn(Zp).
The second construction in this section is the direct product, which takes two known
groups and constructs a new one, using ordered pairs. This can be extended to n-tuples,
where the entry in the ith component comes from a group Gi, and n-tuples are multiplied
component-by-component. This generalizes the construction of n-dimensional vector spaces
(that case is much simpler since every entry comes from the same set, and the same operation
is used in each component).
SOLVED PROBLEMS: §3.3
19. Show that Z5 × Z3 is a cyclic group, and list all of the generators of the group.
20. Find the order of the element ([9]12, [15]18) in the group Z12 × Z18.
21. Find two groups G1 and G2 whose direct product G1 × G2 has a subgroup that is not
of the form H1 × H2, for subgroups H1 ⊆ G1 and H2 ⊆ G2.
22. In the group G = Z × 36 , let H = {[x] | x ≡ 1 (mod 4)} and K = {[y] | y ≡ 1 (mod 9)}.
Show that H and K are subgroups of G, and find the subgroup HK.
23. Let F be a field, and let H be the subset of GL2(F ) consisting of all upper triangular
matrices. Show that H is a subgroup of GL2(F ).
24. Let p be a prime number.
(a) Show that the order of the general linear group GL2(Zp) is (p 2 − 1)(p 2 − p).
Hint: Count the number of ways to construct two linearly independent rows.
(b) Show that the order of the subgroup of upper triangular matrices is (p − 1) 2 p.