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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?