Isomorphism theorems

Nebe, G.

doi:10.1107/97809553602060000541

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.5, p. 34 | 1 | 2 |

Section 1.5.3.5. Isomorphism theorems

Gabriele Nebe^a ^*

^a Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

1.5.3.5. Isomorphism theorems

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[cf. Ledermann (1976), pp. 68–73.]

Remark . If $[\varphi]$ is a homomorphism $[{\cal G} \rightarrow {\cal H}]$ and $[{{\cal N}}]$ $[{\underline{\triangleleft}}]$ $[{{\cal H}}]$ is a normal subgroup of $[{\cal H}]$ , then the pre-image $[\varphi ^{-1} ({{\cal N}}): =]$ $[\{ {\sf g}\in {{\cal G}} \mid]$ $[\varphi({\sf g}) \in {{\cal N}} \}]$ is a normal subgroup of $[{\cal G}]$ . In particular, it holds that $[\ker (\varphi)\,\, {\underline{\triangleleft}}\,\,{{\cal G}}]$ .

Hence the factor group $[{{\cal G}} / \ker(\varphi)]$ is a well defined group. The following theorem says that this group is isomorphic to the image $[\varphi({{\cal G}}) \leq {{\cal H}}]$ of $[\varphi ]$ :

Theorem 1.5.3.5.1. (First isomorphism theorem.) Let $[\varphi: {{\cal G}}\rightarrow {{\cal H}}]$ be a homomorphism of groups. Then $[\overline{\varphi }: {{\cal G}}/\ker(\varphi) \rightarrow \varphi({{\cal G}}) \leq {{\cal H}}.]$ $[{\sf g} \ker(\varphi) \mapsto\varphi({\sf g}) ]$ is an isomorphism between the factor group $[{{\cal G}}/\ker(\varphi)]$ and the image group of $[\varphi]$ , which is a subgroup of $[{\cal H}]$ .

Theorem 1.5.3.5.2. (Third isomorphism theorem.) Let $[{{\cal N}}\, {\underline{\triangleleft}}\,\,{{\cal G}}]$ be a normal subgroup of the group $[{\cal G}]$ and $[{\cal U}\leq {\cal G}]$ be an arbitrary subgroup of $[{\cal G}]$ . Then $[{{\cal U}}\cap {{\cal N}} \,{\underline{\triangleleft}}\,\,{{\cal U}}]$ is a normal subgroup of $[{\cal U}]$ and $[{{\cal U}}/({{\cal U}}\cap {{\cal N}}) \cong {{\cal N}}{{\cal U}} / {{\cal N}}.]$ (For the definition of the group $[{\cal N}]$ $[{\cal U}]$ see Proposition 1.5.3.2.11.)

Definition 1.5.3.5.3. A subgroup $[{{\cal U}}\leq {{\cal H}}]$ is a characteristic subgroup $[{{\cal U}} \, {\rm char} \, {{\cal H}}]$ if $[\varphi ({{\cal U}}) = {{\cal U}}]$ for all automorphisms $[\varphi ]$ of $[{{\cal H}}]$ .

Remarks

(a) If $[{\cal H}]$ is a finite Abelian group and $[{\cal P}]$ is a Sylow p-subgroup of $[{\cal H}]$ , then $[{{\cal P}} \, {\rm char} \, {{\cal H}}]$ , because $[{\cal P}]$ is the only subgroup of $[{\cal H}]$ of order $[|{\cal P}|]$ .
(b) If $[{\cal H}]$ is any group and $[{{\cal U}}\, {\rm char} \, {{\cal H}}]$ , then $[{{\cal U}}\, {\underline{\triangleleft}}\, {\cal H}]$ is also a normal subgroup of $[{\cal H}]$ : for $[{\sf h}\in {\cal H}]$ define the mapping $[\kappa _{{\sf h}}: {\cal H}\rightarrow {\cal H}, {\sf x}\mapsto]$ $[{\sf h} {\sf x} {\sf h}^{-1}]$ . Then $[\kappa _{\sf h}]$ is an automorphism of $[{\cal H}]$ and $[\kappa_{{\sf h}}({{\cal U}}) ={\sf h}\kern2pt {{\cal U}} {\sf h}^{-1} =]$ $[ {{\cal U}}]$ since $[{\cal U}]$ is characteristic in $[ {\cal H}]$ .
(c) If $[{{\cal U}}\, {\rm char} \, {{\cal N}} \, {\underline{\triangleleft}} \, {{\cal H}}]$ , then $[{{\cal U}} \, {\underline{\triangleleft}}\, {{\cal H}}]$ , since the conjugation with any element of $[{\cal H}]$ induces an automorphism of $[{\cal N}]$ .

References

Ledermann, W. (1976). Introduction to group theory. London: Longman. (German: Einführung in die Gruppentheorie, Braunschweig: Vieweg, 1977.)Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, p. 34