Subgroups

Let G be a group and H any subset of G. Let a,b be any two elements of H. Now a,b being members of G the product of a\,b surely belongs to G, but it may or may not belongs to H. If, however, a\,b belongs to H, we say that H is stable for the composition in G and the composition in G has induced the composition in H. If H is itself a group for the induced composition, then we say that H is a subgroup of G.

Definition: A non-empty subset H of a group G is said to be subgroup of G if the composition in G induces a composition in H and if H is a group for the induced composition.

The two subgroups (i) consisting of the identity element alone, and (ii) the group G itself are always present in a group G.

There is, however, trivial subgroup. A subgroup other than these two is known as proper subgroups.

A complex is any subset of a group, whether it is a subgroup or not. It is easy to prove that

(i) The identity of a subgroup is the same as that of a group.

(ii) The inverse of any element of a subgroup is the same as the inverse element regarded as a member of the group.

(iii) The order of any element of a subgroup is the same as that of the element regarded as a member of the group.

Examples:

(i) The additive group of integers is a subgroup of the additive group of rational numbers.

(ii) The multiplicative group of positive rational numbers is a subgroup of the multiplicative group of non-zero real numbers.

(iii) The multiplicative group \left\{ {1, - 1} \right\}is a subgroup of the multiplicative group \left\{ {1, - 1,i, -  i} \right\}.

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