Abstract
In group theory it is a familiar observation that normality of subgroups is not in general a transitive relation, i.e., H ⊲ K ⊲ G need not imply that H ⊲ G. The smallest group exhibiting this phenomenon is Dih(8), the dihedral group of order 8.
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References
Baer, R. Situation der Untergruppen und Struktur der Gruppe, Sitz.-Ber. Heidelberg. Akad. Wiss. 2, 12–17, 1933.
Best, E. and Taussky, O. A class of groups, Proc. Roy. Irish Acad. Sect A 47, 55–62, 1942.
Carter, R.W. Simple Groups of Lie Type, Wiley-Interscience, New York, 1972.
Cossey, J. Finite groups generated by subnormal J-subgroups, Glasgow Math. J. 37, 363–371, 1995.
Dedekind, R. Über Gruppen, deren sämmtliche Teiler Normalteiler sind, Math. Ann. 48, 548–561, 1897.
Gaschütz, W. Gruppen, in denen das Normalteilersein transitiv ist, J. reine angew. Math. 198, 87–92, 1957.
Griess, R.L. Schur multipliers of the known finite simple groups II, Proc. Symp. Pure Math. 37, 279–282, 1980.
Iwasawa, K. Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Fac. Sci. Imp. Univ. Tokyo I.4, 171–199, 1941.
—, On the structure of infinite M-groups, Jap. J. Math. 18, 709–728, 1943.
Menegazzo, F. Gruppi nei quali la relazione di quasi-normalità è transitiva I, II, Rend. Sem. Mat. Univ. Padova 40, 347–361, 1968; ibid. 42, 389–399, 1970.
Ore, O. Contributions to the theory of groups of finite order, Duke Math 5, 431–460, 1939.
Peng, T.A. Finite groups with pronormal subgroups, Proc. Amer. Math. Soc. 20, 232–234, 1969.
Robinson, D.J.S. Groups in which normality is a transitive relation, Proc. Camb. Phil Soc. 60, 21–38, 1964.
—, A note on finite groups in which normality is transitive, Proc. Amer. Math. Soc. 19, 933–937, 1968.
—, Groups which are minimal with respect to normality being intransitive, Pacific J. Math. 31, 777–785, 1969.
—, Groups whose homomorphic images have a transitive normality relation, Trans. Amer. Math. Soc. 176, 181–213, 1973.
—, A Course in the Theory of Groups, Second Edition, Springer, New York, 1996.
Schmidt, R. Subgroup Lattices of Groups, De Gruyter, Berlin, 1994.
Stammbach, U. Homology in Group Theory, LNM., 359, Springer, Berlin, 1973.
Steinberg, R. Automorphisms of linear groups, Canad. J. Math. 12, 606–615, 1960.
Stonehewer, S.E. Permutable subgroups of infinite groups, Math. Z. 125, 1–16, 1972.
Zacher, G. Caratterizzazione dei t-gruppi risolubili, Ricerche Mat. 1, 287–294, 1952.
—, I gruppi risolubili in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali, Lincei Rend Sc. Fis. Mat. Nat. 37, 150–154, 1964.
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© 1999 Hindustan Book Agency (India) and Indian National Science Academy
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Robinson, D.J.S. (1999). A Survey of Groups in Which Normality or Permutability is a Transitive Relation. In: Passi, I.B.S. (eds) Algebra. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-94-6_11
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DOI: https://doi.org/10.1007/978-93-80250-94-6_11
Publisher Name: Hindustan Book Agency, Gurgaon
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