Transposition (as an element of a symmetric group)

In a sym­met­ric group, a trans­po­si­tion is a per­mu­ta­tion which has the effect of swap­ping two el­e­ments while leav­ing ev­ery­thing else un­changed. More for­mally, it is a per­mu­ta­tion of or­der \(2\) which fixes all but two el­e­ments.

knows-req­ui­site(Cy­cle type of a per­mu­ta­tion): A trans­po­si­tion is pre­cisely an el­e­ment with cy­cle type \(2\).

Example

In \(S_5\), the per­mu­ta­tion \((12)\) is a trans­po­si­tion: it swaps \(1\) and \(2\) while leav­ing all three of the el­e­ments \(3,4,5\) un­changed. How­ever, the per­mu­ta­tion \((124)\) is not a trans­po­si­tion, be­cause it has or­der \(3\), not or­der \(2\).

Parents:

  • Symmetric group

    The sym­met­ric groups form the fun­da­men­tal link be­tween group the­ory and the no­tion of sym­me­try.