International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1. ch. 1.5, p. 34
Section 1.5.3.5. Isomorphism theorems
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Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany |
[cf. Ledermann (1976), pp. 68–73.]
Remark . If is a homomorphism and is a normal subgroup of , then the pre-image is a normal subgroup of . In particular, it holds that .
Hence the factor group is a well defined group. The following theorem says that this group is isomorphic to the image of :
Theorem 1.5.3.5.1. (First isomorphism theorem.) Let be a homomorphism of groups. Then is an isomorphism between the factor group and the image group of , which is a subgroup of .
Theorem 1.5.3.5.2. (Third isomorphism theorem.) Let be a normal subgroup of the group and be an arbitrary subgroup of . Then is a normal subgroup of and (For the definition of the group see Proposition 1.5.3.2.11.)
Remarks
References
Ledermann, W. (1976). Introduction to group theory. London: Longman. (German: Einführung in die Gruppentheorie, Braunschweig: Vieweg, 1977.)Google Scholar