Rational element

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Definition

An element g in a group G is termed a rational element if, whenever h \in G is such that \langle g \rangle = \langle h \rangle, there is an element x \in G such that xgx^{-1} = h. In other words, g and h are conjugate elements in G.

When G is a finite group, this is equivalent to the following four conditions:

  1. For every character \chi of G over the complex numbers, \chi(g) is a rational number.
  2. For every character \chi of G over the real numbers, \chi(g) is an integer.
  3. For r relatively prime to the order of G, g is conjugate to g^r.
  4. For r relatively prime to the order of g, g is conjugate to g^r.

Relation with other properties

Stronger properties

  • Involution: An involution is an element of order two. Any element of order two is rational.

Weaker properties

Related group properties