Dihedral Group D4/Normal Subgroups/Subgroup Generated by a

Example of Normal Subgroup of the Dihedral Group $D_4$

Let the dihedral group $D_4$ be represented by its group presentation:

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$

The subgroup of $D_4$ generated by $\gen a$ is normal.

Proof

Let $N = \gen a$

First it is noted that as $a^4 = e$:

$N = \set {e, a, a^2, a^3}$

and is cyclic.

The left cosets of $N$:

\(\ds e N\)

\(=\)

\(\ds e \set {e, a, a^2, a^3}\)

\(\ds \)

\(=\)

\(\ds \set {e, a, a^2, a^3}\)

\(\ds \)

\(=\)

\(\ds N\)

\(\ds b N\)

\(=\)

\(\ds b \set {e, a, a^2, a^3}\)

\(\ds \)

\(=\)

\(\ds \set {b, b a, b a^2, b a^3}\)

As $\order {\gen a} = \order {D_4} / 2$ it follows from Subgroup of Index 2 is Normal that $\gen a$ is normal.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Example $7.13$