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Conjugation, Normal Subgroups and Factor Groups

Normal Subgroups

Definition: A subgroup $H$ of a group $G$ is normal if the left cosets and right cosets of $H$ are the same. That is, $H$ is normal if, for all $g\in G$, $gH=Hg$.

Examples

  • Every subgroup of an abelian group is normal.
  • The subgroup of rotations of an n-gon is normal in the Dihedral group $D_{n}$.
  • The group \(\mathrm{SL}_{2}(\mathbb{R})\) of \(2\times 2\) matrices with real entries and determinant one is normal in \(\mathrm{GL}_{2}(\mathbb{R})\).
  • The subgroup \(A_{n}\) of even permutations is normal in the symmetric group \(S_{n}\).
  • The subgroup \(\{-1,1\}\) is normal in the quaternion group \(Q=\{\pm 1, \pm i, \pm j, \pm k\}.\)
  • If $G$ is a group, the center $Z(G)$ is normal in $G$.

Non-examples

  • The subgroup of $D_{n}$ generated by a reflection $s$ is not a normal subgroup.
  • The subgroup of $S_{n}$ generated by a cycle is not a normal subgroup.
  • The subgroup $H$ of \(\mathrm{GL}_{2}(\mathbb{R})\) consisting of matrices of the form:
\[H=\{\left(\begin{matrix} 1 & x \\ 0 & 1 \end{matrix}\right) : x\in\mathbb{R}\}\]

is not normal.

Factor Groups

Definition: Let $G$ be a group and $H$ be a normal subgroup. Let $G/H$ be the set of left cosets of $H$ in $G$. Introduce a multiplication on cosets by the rule \((aH)(bH) = abH.\)

Proposition: $G/H$, with the operation described above, is a group, called the “quotient group” or “factor group” of $G$ by $H$.

Key points in the proof:

  • The operation is well defined, meaning that if $aH=a’H$ and $bH=b’H$ then $abH=a’b’H$. This is where the normal hypothesis is needed.
  • The coset $H$ is the identity element.
  • The inverse of $aH$ is $a^{-1}H$.

Note that \(G/H\) has \([G:H]=\frac{\vert G\vert}{\vert H\vert}\) elements.

Examples

  • If $G=\mathbb{Z}$ and $H=n\mathbb{Z}$ then $G/H=\mathbb{Z}_{n}$.
  • If \(G=\mathbb{Z}_{n}\) and \(H=d\mathbb{Z}_{n}\) where $d$ is a divisor of $n$ then \(G/H=\mathbb{Z}_{d}.\)
  • If \(G=\mathrm{GL}_{2}(\mathbb{R})\) and \(H=\mathrm{SL}_{2}(\mathbb{R})\) then \(G/H=\mathbb{R}^{\star}\).
  • If \(G=D_{4}\) and \(H=Z(G)\) then \(G/H\) is isomorphic to \(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\).
  • If \(G=D_{n}\) and \(H\) is the subgroup of rotations, then \(G/H\) is isomorphic to \(\mathbb{Z}_{2}\).

Simple groups

Definition: A group $G$ is simple if it has no nontrivial normal subgroups.

Theorem: \(A_{n}\) is simple if and only if \(n\ge 5\).