Question

The subgroup relation ≤ on the set of subgroups G is reflexive, transitive, and anti-symmetric.

The subgroup relation ≤ on the set of subgroups G is reflexive, transitive, and anti-symmetric.

Homework Answers

Answer #1

Let H be a subgroup of G. Now, every group is a subgroup of itself. Hence, . Hence the subgroup relation is reflexive..

Now, let H1, H2, H3 be three subgroups of G such that . Now, since H1 is a subgroup of H2, and H2 is a subgroup of H3, therefore, H1 must be a subgroup of H3. That is, . Therefore, the subgroup relation is transitive.

Now, let . That is, H1 must also be a subset of H2 and H2 must be a subset of H1. This is only possible if H1=H2. Hence, the subgroup relation is anti-symmetric.

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