On commuting elements of a group

A. I. Zabarina   ; U. A. Guselnikova   ; E. A. Fomina

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2015), pp. 27-32

Résumé

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 $|W|=|U|$; $G\cong D(H)$.

Détail

Mots-clés : group, conjugate element
Keywords: involution, commuting element, generalized dihedral group.

Bibliographie

[1] Chetvertaya studencheskaya olimpiada po algebre Moskovskogo gosudarstvennogo universiteta (2 dekabrya 2009) (in Russian)

[2] Belonogov V. A., Zadachnik po teorii grupp, Nauka Publ., M., 2000, 240 pp. (in Russian)