Subgroup

2 Sub group
Definition 2.1 : Let H be a nonempty subset of the group G. Then H is
said to be a subgroup of G if H is Itself a group under the same operation
as that of G.
Theorem2.2: Let G be a group and . Then H is a subgroup of
G if and only if implies .
Proof: If H is a subgroup and , then and so by
closure condition.
Conversely, suppose that H is a nonempty subset of G that contains the
element whenever .
, then .
Again, for all , by hypothesis .
Now, ( ) . Because H inherits the associated law as a
subset of G, all the group axioms are sasisfied, and therefore a subgroup
of G.
Theorem 2.3: Let H be a finite (nonempty) subset of the group G. If H is
closed under multiplication, then H is a subgroup of G.
Proof: It is suffices to show that whenever . Now, by
closure condition lie in H and, since H is finite cannot
be distinct. Suppose that for . Canceling , we get
, or ( ) ( ) . Thus, .
Corollary 2.4: A nonempty subset H of a finite group G is a subgroup of G
if and only if implies that .
Theorem 2.5: If * + is an indexed collection of subgroups of the group