# 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 $$H$$ is a subset satisfying (b) and (c). Pick any $$h \in H$$. By (b), we have $$h^{-1} \in H$$ and so by (c) we have $$h * h^{-1} \in H$$, which gives us $$e \in H$$.

Note by Calvin Lin
4 years, 11 months ago

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

1. $$e \in H$$

2. if $$g,h \in H$$, then $$gh^{-1} \in H$$

or

1. $$H \neq \varnothing$$

2. if $$g,h \in H$$, then $$gh^{-1} \in H$$

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

- 4 years, 10 months ago

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

- 4 years, 10 months ago