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Computing normal subgroups Alexander Hulpke School of Mathematical and Computational Sciences University of St Andrews The North Haugh St Andrews Fife KY16 9SS United Kingdom ahulpke dcs st and ac uk http www gap dcs st and ac uk ahulpke intersection of conjugates in a composition series of G this yields a series of normal subgroups and refining elementary abelian subfactors using the MeatAxe 18 For permutation groups alternatively the composition series algorithm can be modified to compute normal subgroups in the first place 3 Using the homomorphism principle we compute the normal subgroups of G inductively along this series in a step from Ni to Ni 1 Ni computing the normal subgroups of G Ni 1 from those of G Ni After the last step for Ni 1 h1i this yields the normal subgroups of G Abstract This note presents a new algorithm for the computation of the set of normal subgroups of a finite group It is based on lifting via homomorphic images 1 Introduction Next to a composition series the structure of a finite group is reflected by the lattice of its normal subgroups It indicates for example possible epimorphic images or decompositions as sub direct products Furthermore it contains all characteristic subgroups Contrasting this structural prominence however the problem never seems to have been considered seriously in the literature and the best algorithm known to the author is the na ve approach to take normal closures of conjugacy classes This requires a priori knowledge of the conjugacy classes The alternative approach presented here avoids this requirement and performs much quicker for larger groups even if the classes are known for free This will permit to compute the normal subgroups in reasonable time even for fairly large groups for which for example the conjugacy classes could not be computed Evidence for this is given in the last section The algorithm described below does not explicitly refer to a groups representation but requires several auxiliary routines



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