Explore BrainMass

Explore BrainMass

    Non-Empty Subsets of a Group

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Let G be a group and H be a non-empty subset of G. We define H^2 as the set of elements of G which can be written in the form h_1 h_2, with h_1 ,h_2∈H.
    Let H be finite. Prove that H is a subgroup if and only if H^2=H.

    © BrainMass Inc. brainmass.com October 10, 2019, 6:51 am ad1c9bdddf

    Solution Preview

    Proof. Assume that H is a subgroup. Then it is closed under multiplication, and hence H^2 is a subset of H. For the converse, let us assume that h∈H. Since h=he, and e∈H, we conclude that H is a subset of H^2, and hence ...

    Solution Summary

    The solution provides the criterion for a finite non-empty subset of a group to be a subgroup.