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 c J. Proposition 2. 1) 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: VuеU VwеW VgеG (u gеU w gеW). 2) A product of two trivially commuting elements is a non-trivially commuting element: u1 е U, ^е U UlU2е 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| * 0, then |U| = | W|. Corollary 4. Let |G| = n, |U| * 0, then 1) Vwе WVu еU3u'', u''^U(w = u u'' = u"u ); 2) U = J; 3) |G| = n = 4q + 2. Theorem 5. Let |G| = n, |U| * 0, then W is a commutative normal divisor of the group G. Proposition 6. Let (А, •) be an Abelian group with the involution and ф(А), ◦) be a generalized dihedral group. Then the set U of trivial commuting elements of the group D^) is empty. Theorem 7. Let ф(А), ◦) be a generalized dihedral group and let the group А have no involutions. Then the set U of trivially commuting elements of the group D^) is the set {(a, -1)| aеA}, |U| = W Theorem 8. Let (G, •) be a group, the set of involutions J of the group G be not empty, and the set H = G\J be a subgroup, H * {e}. Then 1) H is a commutative normal divisor of G; |G/H| = 2; 2) The set U of trivially commuting elements of the group G coincides with J and |W| = |U|; 3) G = D(H).