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Math and science::Algebra::Aluffi

Subgroups. Definition.

Subgroup

Let (G,mG) be a group. A group (H,mH) is a subgroup of G iff both of the following conditions hold:

  1. H⊆G
  2. The inclusion function i:H→G [has what property?].

An alternative form of the subgroup condition is the following pair of conditions:

  1. ∀g∈G,g∈H⟹g−1∈H.
  2. ∀g1,g2∈G,g1,g2∈H⟹mG(g1,g2)∈H.

In words, these two conditions say:

  1. For all elements of H, [what?].
  2. H is [what?].

Aluffi has a condensed formula that combines these two conditions into one. Can you remember it?