Exponent of semidirect product may be strictly greater than lcm of exponents

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Statement

In terms of internal semidirect products

It is possible to have a finite group G, a complemented normal subgroup N of G with a complement H (so G is an internal semidirect product of N and H) such that the exponent of G is strictly greater than the lcm of the exponents of N and H.

Related facts

Proof

Example of dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Consider G to be the dihedral group:D8:

G = \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle

This has eight elements:

\! G = \{ e,a,a^2,a^3,x,ax,a^2x,a^3x\}

Suppose N is one of the Klein four-subgroups of dihedral group:D8:

\! N = \{ e, a^2, x, a^2x \}

and take H as one of the non-normal subgroups of dihedral group:D8 that is not contained in N:

\! H = \{ e, ax \}

We note that:

  • The exponent of G is 4.
  • The exponent of N is 2.
  • The exponent of H is 2.

Thus, the exponent of G is strictly greater than the lcm of the exponents of N and H.