Lagrange's Theorem

This week, we continue our study of Group Theory with a guest post by Chu-Wee Lim on Lagrange's Theorem.

You may first choose to read the post on Group Theory if you have not already done so.

Is the following proof correct?

Problem: Show that in the definition of a subgroup, conditions (b) and (c) imply (a).

Proof: Suppose HH is a subset satisfying (b) and (c). Pick any h∈H h \in H . By (b), we have h−1∈H h^{-1} \in H and so by (c) we have h∗h−1∈Hh * h^{-1} \in H , which gives us e∈H e \in H .

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Note by Calvin Lin
7 years, 9 months ago

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Comments

You can cut the number of conditions for a subgroup down to two. Either

  1. e∈He \in H

  2. if g,h∈Hg,h \in H, then gh−1∈Hgh^{-1} \in H

or

  1. H≠∅H \neq \varnothing

  2. if g,h∈Hg,h \in H, then gh−1∈Hgh^{-1} \in H

Either way around, you need to have a condition which guarantees the existence of elements in HH. You can conflate conditions (b) and (c), but you can't drop (a).

Mark Hennings - 7 years, 9 months ago

The assumption that you can pick an h∈Hh \in H is wrong. Only having conditions (b) and (c) would allow the empty subset to be a subgroup.

Zef RosnBrick - 7 years, 9 months ago
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