Constructing normalisers in finite soluble groups

AbstractThis paper describes algorithms for constructing a Hall π-subgroup H of a finite soluble group G and the normaliser NG(H). If G has composition length n, then H and NG(H) can be constructed using O(n4 log |G|) and O(n5 log |G|) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative system normalisers. Computer implementations of these algorithms can compute a Sylow 3-subgroup of a group with n = 84, and its normaliser in 47 seconds and 30 seconds, respectively. Constructing normalisers of arbitrary subgroups of a finite soluble group can be complicated. This is shown by an example where constructing a normaliser is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms