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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, 3 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).

Mark Hennings - 4 years, 3 months ago

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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.

Zef RosnBrick - 4 years, 3 months ago

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