2 burnside

Another application (known to the OP, but interesting enough to describe clearly) is a result of Burnside (1905) classifying for which powers b = b(p, a, q) there is a group of order paqb with no non-identity normal p-subgroup: the classification is based simply on the orders of the automorphism groups of the p-subgroups. Burnside had an error in his analysis of the associated arithmetical condition that was corrected in Coates–Dwan–Rose (1976). Burnside's result was generalized in Glauberman (1975) and Bialostocki (1975, 1987). Many of these and further results are based on analyzing nilpotent p-subgroups of GL(n, p), resting ultimately on the fact that that every p-subgroup of the automorphism group of a p-group of rank n is isomorphic (including in some sense, its action) to a subgroup of GL(n, p).

• Burnside, W.On groups of order pαqβLond. M. S. Proc. (2) 1, 388-392 (1904).JFM35.0162.01DOI:10.1112/plms/s2-1.1.388

• Burnside, W.On groups of order pαqβ (second paper).Lond. M. S. Proc. (ser 2) 2, (1905) 432-437.JFM36.0198.02DOI:10.1112/plms/s2-2.1.432

• Glauberman, G.On Burnside's other paqb theorem.Pacific J. Math. 56 (1975), no. 2, 469–476.MR412269URL:euclid.pjm/1102906371

• Bialostocki, Arie.On products of two nilpotent subgroups of a finite group.Israel J. Math. 20 (1975), no. 2, 178–188.MR407148DOI:10.1007/BF02757885

• Coates, Martin; Dwan, Michael; Rose, John S.A note on Burnside's other pαqβ theorem.J. London Math. Soc. (2) 14 (1976), no. 1, 160–166.MR419594DOI:10.1112/jlms/s2-14.1.160

• Bialostocki, Arie.On the other pαqβ theorem of Burnside.Groups–St. Andrews 1985.Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 1, 41–49.MR879428DOI:10.1017/S0013091500017946

• O'Brien, E. A.(5-ANUM)

• 1

This is just Burnside's basis theorem. See for instance theorem 12.2.2 on page 178 of M. Hall, Jr.'s textbook on the Theory of Groups. The original reference for the phrasing in terms of automorphisms is from P. Hall (1933).

As far as where to go from it, this is roughly how automorphism groups of p-groups are calcuated in O'Brien (1992), Eick–Leedham-Green–O'Brien (2002) and the AutPGrp package of GAP. This is often useful in understand fusion systems, where the p-core of automorphism groups is under good control, and so the GL(n, p) part is the primary interest.

• Hall, P. A contribution to the theory of groups of prime-power order. Proc. Lond. Math. Soc., Ser. 2, 36, (1933) 29-95. Zbl0007.29102 DOI:10.1112/plms/s2-36.1.29
• O'Brien, E. A.(5-ANUM) Computing automorphism groups of p-groups. Computational algebra and number theory (Sydney, 1992), 83–90, Math. Appl., 325, Kluwer Acad. Publ., Dordrecht, 1995. MR1344923
• Eick, Bettina; Leedham-Green, C. R.; O'Brien, E. A. Constructing automorphism groups of p-groups. Comm. Algebra 30 (2002), no. 5, 2271–2295. MR1904637 DOI10.1081/AGB-120003468