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

28 CHAPTER 3. GROUPS
47. Let T be the set of functions tc,d : R 2 → R 2 defined by tc,d(x1, x2) = (x1 + c, x2 + d),
where c, d are any elements of R. Show that T is a group under composition of
functions.
48. Let G be a group. For x, y ∈ G, define x ∼ y if there exists some element a ∈ G such
that y = axa −1 . Show that ∼ defines an equivalence relation on G.
Note: This is a general case of Exercise 2.3.15 in the text.
3.2 Subgroups
Many times a group is defined by looking at a subset of a known group. If the subset is
a group in its own right, using the same operation as the larger set, then it is called a
subgroup. For instance, any group of permutations is a subgroup of Sym(S), for some set
S. Any group of n × n matrices (with entries in R) is a subgroup of GLn(R).
If the idea of a subgroup reminds you of studying subspaces in your linear algebra course,
you are right. If you only look at the operation of addition in a vector space, it forms an
abelian group, and any subspace is automatically a subgroup. Now might be a good time
to pick up your linear algebra text and review vector spaces and subspaces.
Lagrange’s theorem (Theorem 3.2.10) is a very important result. It states that in a
finite group the number of elements in any subgroup must be a divisor of the total number
of elements in the group. This is a useful fact to know when you are looking for subgroups
in a given group. In particular, any group of prime order contains no proper nontrivial
subgroups and must therefore be cyclic.
It is also important to remember that every element a in a group defines a subgroup
〈a〉, consisting of all powers (positive and negative) of the element. This subgroup has o(a)
elements, where o(a) is the order of a. If the group is finite, then to find 〈a〉 you only need
to calculate the positive powers of a, since in that case the inverse a−1 of any element can
be expressed in the form an , for some n > 0.
Lagrange’s Theorem implies that the order of any element of a finite group is a divisor
of the order of the group. This can be extremely helpful, as the following example shows.
. Since
|Z × 17
Suppose that we are attempting to show that [2]17 is a generator of the group Z × 17
| = 17, the possible orders of [2] are 2, 4, 8, or 16. We do not have to calculate all
powers of [2] to find its order. We have [2] 2 = [4], [2] 4 = [16] = [−1], so [2] does not have
order 2 or 4, but the [2] 8 = ([2] 4 ) 2 = [−1] 2 = [1] so [2] has order 8, showing that [2] is not
a generator. A similar argument shows that [3] 2 = [9], [3] 4 = [−4], and [3] 8 = [16], so [3] is
a generator of Z × 17 since it must have order 16.
SOLVED PROBLEMS: §3.2
28. In Zn, show that if gcd(a, n) = d, then 〈[a]n〉 = 〈[d]n〉.
Note: This result is very useful when you find cyclic subgroups of Zn.