Minimal extensions as generalized semidirect products

Abstract

An extension of a two‐element symmetry group Z₂ (defined, e.g., by C, P, or T) by another symmetry group K is called a minimal extension (ME). From the general theory of group extensions it follows that a ME is determined by an element φ of K and an automorphism F in K which are related in a certain way. It is shown that every ME can be expressed as a generalized semidirect product $(\mathrm{GSP}{\mathrm{):\hspace{0.17em}(K\hspace{0.17em}[circled\; tau]\hspace{0.17em}H)/K\prime }}_0$ , where the homomorphism τ is defined by F, and K′₀,H are cyclic groups of order m and 2m, respectively, m being the order of φ. The simplest GSP form of any ME is obtained depending on φ being outside or inside the center of K (in particular, φ may be equal to the unit element), and F being an inner or an outer automorphism. A complete classification of inequivalent ME's is presented, and its possible significance for magnetic space and point groups is indicated. The usefulness of the GSP form for finding the irreducible representations of a ME is pointed out.