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- W1986535052 abstract "Suppose M is a normal subgroup of the group G and M is a direct product of subgroups Li, 1 1. The objective of the present paper is to show that the problem of finding the automorphisms of G may be reduced to that of finding certain isomorphisms of the groups G/M and N/K. If n = 1, then N/K= G and our theorems are trivialities. Thus our results are really concerned with the case when n> 1. Suppose SI is an automorphism of G. If M*, N*, and K* are the images under a of M, N, and K, respectively, then a induces an isomorphism /? of G/M onto G/M* and an isomorphism y of NfK onto N*IK*. Further B maps the subgroup N/M onto N*!M*, 7 maps M/K onto M*fK*, and /? and 7 induce the same mapping of N/M onto N*/M*. (Here we identify N/M and N*/M* with (N:K)f( MiK) and (N*/K* )/( M*/K* ), respectively; thus the previous statement simply means that if y(Kn) = K*n* for ne N and n* EN*, then /?(Mn) = M*n*.) The thrust of the present paper is to reverse the above. Namely, suppose fl is an isomorphism of G/M onto G/M* for some normal subgroup M* and suppose y is an isomorphism of N/K onto N*jK* for some subgroups N* and K* with K* a normal subgroup of N*. Assume that K* < M* < N*, fl maps KIM onto N*/M*, y maps M/K onto M*/K*, and /I? and 7 induce the same mapping of N/M onto N*/M*. If certain other hypotheses are satisfied (the other assumptions basically stating that M*, N*, and K* are “like” M, N, and K), we show that there is an automorphism u of G such that z induces both /I’ and 7." @default.
- W1986535052 created "2016-06-24" @default.
- W1986535052 creator A5008437847 @default.
- W1986535052 date "1987-01-01" @default.
- W1986535052 modified "2023-09-27" @default.
- W1986535052 title "Automorphisms of induced extensions" @default.
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- W1986535052 doi "https://doi.org/10.1007/bfb0078687" @default.
- W1986535052 hasPublicationYear "1987" @default.
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