Subgroup

A subgroup of a group ${\displaystyle (G,\cdot )}$ is any subset ${\displaystyle H}$ that is also a group under the same operation. Equivalently, the inverse of an element, or the product of two elements in ${\displaystyle H}$, must remain in ${\displaystyle H}$. More concisely, a subgroup of a group is a subset closed under the group product and inverses. When this happens, we can write ${\displaystyle H\leq G}$.

Every nontrivial group ${\displaystyle (G,\cdot )}$ contains at least two subgroups: the trivial subgroup ${\displaystyle \{1\}}$, and the entire group ${\displaystyle G}$. With the exception of cyclic groups with prime order, there will always be some other subgroup.

Subgroups are closed under arbitrary intersection, which means we can talk about the smallest subgroup containing a given set ${\displaystyle S}$. This is called the subgroup generated by ${\displaystyle S}$, and is denoted ${\displaystyle \langle S\rangle }$. If ${\displaystyle S}$ consists of finitely many elements ${\displaystyle a_{1},\ldots ,a_{n}}$, we can also write ${\displaystyle \langle a_{1},\ldots ,a_{n}\rangle }$ for this same subgroup. This notation should not be confused with that of group presentations.

In fact, the subgroups of any group form a complete lattice.

Examples

• The multiples of any integer ${\displaystyle n}$ form a subgroup of the group of integers under addition ${\displaystyle \mathbb {Z} }$. This can be denoted by ${\displaystyle n\mathbb {Z} }$.
• Tetrahedral symmetry is a subgroup of octahedral symmetry, as a regular tetrahedron can be inscribed symmetrically on the vertices of a cube (see stella octangula).
• If ${\displaystyle m\leq n}$, the symmetric group ${\displaystyle S_{m}}$ can be seen as a subgroup of ${\displaystyle S_{n}}$, by considering the permutations that fix all but the first ${\displaystyle m}$ elements.

Cosets

Suppose ${\displaystyle H}$ is a subgroup of ${\displaystyle G}$. The left cosets of ${\displaystyle H}$ in ${\displaystyle G}$ are the sets of the form

${\displaystyle gH=\{gh:h\in H\}.}$

The right cosets ${\displaystyle Hg}$ are defined in an entirely analogous way. Note that ${\displaystyle H}$ itself will always be both a left and right coset. Cosets are not to be confused with the subgroups themselves. The only coset that is also a subgroup is ${\displaystyle H}$.

The key results on cosets are the following.

• Any two cosets, either left or right, are in bijection and thus have as many elements as ${\displaystyle H}$.
• Left cosets partition ${\displaystyle G}$, as do right cosets.

An immediate corollary is Lagrange's theorem: if ${\displaystyle H}$ is a subgroup of ${\displaystyle G}$, then the order of ${\displaystyle H}$ divides the order of ${\displaystyle G}$. The value ${\displaystyle |G|/|H|}$ can be written ${\displaystyle [G:H]}$ and is called the index of the subgroup.

Normal subgroups

In general, the left and right cosets of a subgroup won't be the same. If this happens, the subgroup is called normal, and we write HG. Normal subgroups are of key importance within group theory, as they make it possible to define group quotients. Note that every subgroup of an abelian group is normal.

A subgroup ${\displaystyle H\leq G}$ is normal if and only if ${\displaystyle gH=Hg}$ for any ${\displaystyle g\in G}$. Equivalently, ${\displaystyle g^{-1}hg\in H}$ for any ${\displaystyle g\in G}$, ${\displaystyle h\in H}$.

Just as all subgroups do, normal subgroups are closed under arbitrary intersection, and thus form a complete lattice.

A group without any nontrivial normal subgroups is called simple. As mentioned earlier, cyclic groups of prime order have no nontrivial subgroups, and are thus simple. Moreover, it can be proved that any alternating group of degree at least 5 is simple. The complete classification of finite simple groups was a long-standing problem in mathematics, only fully solved at the beginning of the 21st century, when these groups were classified into 4 infinite families (two of which were just mentioned), plus 26 exceptional cases.