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Math and science::Algebra::Aluffi

Subgroups. Definition.

Subgroup

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

  1. \( H \subseteq G \)
  2. The inclusion function \( i: H \to G \) [has what property?].

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

  1. \( \forall g \in G, \; g \in H \implies g^{-1} \in H \).
  2. \( \forall g_1, g_2 \in G, \; g_1, g_2 \in H \implies m_G(g_1, g_2) \in H \).

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?