Subgroups
A natural question to ask is whether we can have a smaller group inside of a particular group. We begin by saying that S is a subset of a group G, denoted S ⊆ G, if S consists only of the elements of G. The empty set { } is always considered to be a subset, but we will restrict our attention to non-empty subsets.
Definition 7.2.1. A subgroup S of a group G is a non-empty subset of G which is itself a group under the same operation as G.
When proving that a subset of a group is a subgroup, it is never necessary to check associativity. Since the associative law holds for all elements of the group, it automatically holds when the elements are in some subset H.
To determine whether a subset H of a group G with an operation ∗ is a subgroup, we need not verify all the axioms. The following axioms (S1) and (S2) are sufficient.
Theorem 7.2.2. A nonempty subset H of a group G is a subgroup of G provided that
(S1) if a, b ∈ H, then ab ∈ H;
(S2) if a ∈ H, then a−1 ∈ H.
Properties (S1) and (S2) are the closure and inverse axioms for a group. Associativity holds in H, as noted above. Thus we need only verify that e ∈ H. Since H is nonempty, there exists an element c ∈ H. By (S2), c−1 ∈ H, and by (S1) cc−1 ∈ e is in H. Therefore H is a group. □.
The subset {e} that contains the identity element e of G alone forms a subgroup of G. G may be considered as a subgroup of itself. These two subgroups are trivial subgroups of G. The set of all integers forms a subgroup of the group of all rational numbers under real addition +.
Remark. S = {−1, 1} is a nonempty subset of ℚ, and it is a group with respect to multiplication that is a subgroup (ℚ,*), but it is not a subgroup of (ℚ,+).
Every group G automatically has two obvious subgroups, namely G itself and the subgroup consisting of the identity element, e, alone. These two subgroups we call trivial subgroups. Our interest will be in the remaining ones, the proper subgroups of G.
Subgroup S of a group G is indicated with the following notations
H ≤ G, or H < G (if H ≠ G)
7.2.3 Definition. Let (G, ⋅) a group. The center of G is the subset of G consinsting of all those elements that commute with every element of G i.e.
Z(G) = {g ∈ G | gx = xg ∀x ∈ G}
The notation Z(G) comes from the fact that the German word for center is Zentrum. The term was coined by J. A. de Séguier in 1904. Note that the centre of G is equal to G if and only if G is abelian.
7.2.4 Proposition. The center of a group G is a subgroup of G.
Proof. Clearly, e ∈ Z(G), so Z(G) is nonempty. Now, suppose a, b ∈ Z(G). Then (ab)x = a(bx) = a(xb) = (ax)b = (xa)b = x(ab) for all x ∈ G; and, therefore, ab ∈ Z(G), so that Z(G) satisfies the first axiom (associativity) for groups. Furthermore, multiplying both sides of the equation xg = gx first on the right side then on the left by x−1, we deduce that gx−1 = x−1g, which shows that x−1 ∈ Z(G). Satisfying all axioms, we conclude Z(G) is a subgroup of G. □