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Product of conjugates is proper

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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This article describes a result of the form that argues that a subset constructed in a certain fashion is proper, viz it is not the whole group


Contents

Statement

Verbal statement

Given any two proper subgroups of a group that are conjugate to each other, their product is a proper subset of the group.

Symbolic statement

Let H \le G be a proper subgroup, and let H^g = g^{-1}Hg be a conjugate of H. Then HH^g is a proper subset of G.

Related facts

Applications and similar facts

Proof

Given: A finite group G, two proper conjugate subgroups H and H^g, where H^g = g^{-1}Hg.

To prove: HH^g is a proper subset of G.

Proof: Suppose not, i.e., suppose HH^g = G.

  1. g \in HH^g, so we can write g = hk where h \in H, k \in H^g.
  2. Thus, H^g = H^{hk} = H^k. This yields H = (H^g)^{k^{-1}}.
  3. But we know that k \in H^g, so we get H = H^g.
  4. We thus get G = HH^g = H, contradicting the assumption that H is a proper subgroup of G.