25 The Lattice of Subgroups of a Group

#todo/school : Do problem 7 📅 2025-02-02 ✅ 2025-02-02

7

Question

Find the Center of $D_{16}$ . (see also Dihedral Groups)

Proof

Here:

D_{16} = ⟨ r, s | r^{8} = 1, s^{2} = 1, r s = s r^{- 1} ⟩

We want to find:

Z (D_{16}) = C_{D_{16}} (D_{16}) = {g \in D_{16} | \forall x \in D_{16} (g x = x g)}

We want to find elements in $D_{16}$ that don't commute. We only have to check elements that are in each Centralizers of $D_{16}$ . We only have to care about the generators and their various powers (up to Order of course).

$1$ will commute with everything so $1 \in Z (D_{16})$ .
$r$ does not commute with $s$ since $s r \neq r s$
$r^{2}$ does not commute with $s$ since $s r^{2} \neq r^{2} s$
...
$r^{4}$ does commute with $s$ since $r^{4} s = s r^{4}$ . Further with any $r^{i} s$ then $r^{4} r^{i} s = r^{i} s r^{4}$ since
...
$r^{7}$ doesn't commute with $s$
$s$ doesn't commute with $r$ .

Thus we have $Z (D_{16}) = {1, r^{4}}$ .

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