Same order iff potentially conjugate

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Statement

Verbal statement

The following are equivalent for two elements of a group:

  • They have the same order.
  • The group can be embedded in a bigger group in which the two elements are conjugate elements.

Statement with symbols

The following are equivalent for elements x,y \in G:

  • The order of x equals the order of y.
  • There exists a group L containing G such that x,y are conjugate in L.

Related facts

Applications

Facts used

  1. Isomorphic iff potentially conjugate