Write down a careful prove of the following theorem by provi

Write down a careful prove of the following theorem by proving a. and b. Let G be a group with respect to a binary operation that is written as multiplication. Let H be a nonempty subset of a group. Then H is a subgroup of G if and only if a, b elementof H imply that ab^-1 elementof H a. lf H is a subgroup of G, then a, b elementof H imply that ab^-1 elementof H b.If H be a nonempty subset of a group and a, b elementof H imply that ab^-1 elementof H, then H is a subgroup of G.

Solution

Let G be a group. Show that a nonempty subset H is a subgroup of G if any only if ab¹H for any a,bH.

The forward direction is quite easy. Suppose H is a subgroup. Then by closure, abH for any a,bH.

Every element has an inverse. Hence, if bH, then b¹H. Hence, by closure again, ab¹H.

Backward direction, suppose ab¹H for any a,bH.

Take aa, b in H then c=b¹H, so that ac¹=a(b¹)¹=abH.

thus H is a subgroup of G.