Elementary Group Theory - Subgroups

Subgroups

A subset H of G is called a subgroup of a group <G,*> if H satisfies the axioms of a group, using the same operator "*", and restricted to the subset H. Thus if H is a subgroup of <G,*>, then <H,*> is also a group, and obeys the above theorems, restricted to H. The order of subgroup H is the number of elements in H.

A proper subgroup of a group G is a subgroup which is not identical to G. A non-trivial subgroup of G is (usually) any proper subgroup of G which contains an element other than e.

Theorem 2.1: If H is a subgroup of <G,*>, then the identity e_H in H is identical to the identity e in (G,*).

Proof. If h is in H, then h*e_H = h; since h must also be in G, h*e = h; so by theorem 1.8, e_H = e.

Theorem 2.2: If H is a subgroup of G, and h is an element of H, then the inverse of h in H is identical to the inverse of h in G.

Proof. Let h and k be elements of H, such that h*k = e; since h must also be in G, h*h -1 = e; so by theorem 1.5, k = h -1.

Given a subset S of G, we often want to determine whether or not S is also a subgroup of G. A handy theorem valid for both infinite and finite groups is:

Theorem 2.3: If S is a non-empty subset of G, then S is a subgroup of G if and only if for all a,b in S, a*b -1 is in S.

Proof. If for all a, b in S, a*b -1 is in S, then

• e is in S, since a*a -1 = e is in S.
• for all a in S, e*a -1 = a -1 is in S
• for all a, b in S, a*b = a*(b -1) -1 is in S

Thus, the axioms of closure, identity, and inverses are satisfied, and associativity is inherited; so S is subgroup.

Conversely, if S is a subgroup of G, then it obeys the axioms of a group.

• As noted above, the identity in S is identical to the identity e in G.
• By A4, for all b in S, b -1 is in S
• By A1, a*b -1 is in S.

The intersection of two or more subgroups is again a subgroup.

Theorem 2.4: The intersection of any non-empty set of subgroups of a group G is a subgroup.

Proof. Let {H_i} be a set of subgroups of G, and let K = ∩{H_i}. e is a member of every H_i by theorem 2.1; so K is not empty. If h and k are elements of K, then for all i,

• h and k are in H_i.
• By the previous theorem, h*k -1 is in H_i
• Therefore, h*k -1 is in ∩{H_i}.

Therefore for all h, k in K, h*k -1 is in K. Then by the previous theorem, K=∩{H_i} is a subgroup of G; and in fact K is a subgroup of each H_i.

Given a group <G,*>, define x*x as x², x*x*x*...*x (n times) as xn, and define x0 = e. Similarly, let x -n for (x -1)n. Then we have:

Theorem 2.5: Let a be an element of a group (G,*). Then the set {an: n is an integer} is a subgroup of G.

A subgroup of this type is called a cyclic subgroup; the subgroup of the powers of a is often written as <a>, and we say that a generates <a>.