On commuting elements of a group

Let $G$ be an arbitrary group. An element g is trivially commuting if it does not commute with any other elements but itself and unity. We call an element as non-trivially commuting if it is not trivially commuting. Sets of all trivially commuting elements, non-trivially commuting elements, and involutions of the group are denoted by $U$, $W$, $J$.
Proposition 1. $U\subset J$.
Proposition 2.

• An element conjugate to a trivially commuting element is a trivially commuting element; an element conjugate to a not trivially commuting element is a non-trivially commuting element элемент: $\forall u\in U\ \forall w\in W\ \forall g\in G (u^g\in U\ w^g\in W)$.
• A product of two trivially commuting elements is a non-trivially commuting element: $u_1\in U, u_2\in U\Rightarrow u_1u_2\in W$.

Theorem 3. If the set of trivially commuting elements of a finite group is not empty, they are exactly half to the group: let $|G|=n$, $|U|\ne0$, then $|U|=|W|$.
Corollary 4. Let $|G|= n$, $|U|\ne0$, then

• $\forall w\in W\ \forall u^*\in U\ \exists u', u''\in U (w=u^*u'=u''u^*)$;
• $U=J$;
• $|G|=n= 4q+2$.

Theorem 5. Let $|G|=n$, $|U|\ne0$, then $W$ is a commutative normal divisor of the group $G$.
Proposition 6. Let $\langle A, \cdot\rangle$, be an Abelian group with the involution and $\langle D(A), \circ\rangle$ be a generalized dihedral group. Then the set $U$ of trivial commuting elements of the group $D(A)$ is empty.
Theorem 7. Let $\langle D(A), \circ\rangle$ be a generalized dihedral group and let the group $A$ have no involutions. Then the set $U$ of trivially commuting elements of the group $D(A)$ is the set $\{(a, -1)\mid a\in A\}$, $|U|=|W|$.
Theorem 8. Let $\langle G, \cdot\rangle$ be a group, the set of involutions $J$ of the group $G$ be not empty, and the set $H=G\setminus J$ be a subgroup, $H\ne\{e\}$. Then

• $H$ is a commutative normal divisor of $G$; $|G/H|=2$;
• The set $U$ of trivially commuting elements of the group $G$ coincides with $J$ and $|W|=|U|$;
• $G\cong D(H)$.