If G, * is a group and H is a nonempty infinite subset of G which is closed unde
ID: 3103540 • Letter: I
Question
If G, * is a group and H is a nonempty infinite subset of G which is closed under *, must H be a subgroup of G under the induced operation of * on H? Prove or give a counterexample.Explanation / Answer
True. Let x be an element of H, and use additive notation for the operation, so that H must also contain: x, 2x, 3x, 4x, ... etc. Since H is finite this must repeat at some point. In other words, there exists a>b>0 such that: ax = bx But this means that (1) (a-b)x = 0 i.e. (2) x + (a-b-1)x = 0 (Remember, a and b are just positive integers representing the number of x's to be added. They are NOT given as element of G or H.) We know by assumption H is closed under the operation. (1) shows that indeed 0, the identity, is in H. (2) shows that, given any x in H, the inverse to x (namely (a-b-1)x) is also in H. Therefore H is a subgroup of G under the induced operation.