What is a Subgroup? Definition, Equations & Explanation

[Greg Bernhardt]

Definition/Summary

A subgroup H of a group G is a set of elements of G with G's group operation where H is also a group. The identity of G is also in H. The identity group and G itself are both trivial subgroups of G.

With a subgroup, one can partition a group's elements into left cosets and right cosets, where each side of cosets is disjoint, and where every coset contains the same number of elements as the subgroup. Lagrange's theorem follows:

If G is finite group, then order(H) evenly divides order(G) for every subgroup H.

If a subgroup's left cosets equal its right cosets, then the subgroup is a normal subgroup, and it is self-conjugate.

Equations

Left coset: gH = \{gh : h \in H\}
Right coset: Hg = \{hg : h \in H\}

Conjugate of H by g: H^g = gHg^{-1} = \{ghg^{-1} : h \in H\}

Extended explanation

Proof that a normal subgroup is self-conjugate.

For g in G, left coset gH is equal to right coset Hg, from normality and from both cosets containing g. This means that for every h₁ in H, there is a h₂ in H such that

g*h₁ = h₂*g

Multiplying the right ends of both terms by g^-1 gives

h₂ = g*h₁*g^-1

or H = gHg^-1 -- self-conjugacy.

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[fresh_42]

A good exercise which provides insights is the following:
Prove that a subgroup is normal if and only if its quotient has a group structure.

Given a group $G$ and a subgroup $U<G$. Then we can always consider the set $G/U=\{\,gU\,|\,g\in G\,\}$ of equivalence classes with respect to $U$. But $G/U$ is only a group itself, if $U \triangleleft G$ is a normal subgroup.