# zbMATH — the first resource for mathematics

Actions and automorphisms of crossed modules. (English) Zbl 0719.20018
The reviewer suggested in his paper [in Low Dimensional Topology, Lond. Math. Soc. Lect. Note Ser. 48, 215-238 (1982; Zbl 0484.55007)] that crossed modules could be regarded as ‘2-dimensional groups’. In this direction, the paper under review extends some aspects of the theory of automorphisms of groups to automorphisms of crossed modules. Some of the basic ideas come from J. H. C. Whitehead [Ann. Math., II. Ser. 49, 610-640 (1948; Zbl 0041.101)] and A. S.-T. Lue [Bull. Lond. Math. Soc. 11, 8-16 (1979; Zbl 0416.20030)].
Section 1 defines the actor $${\mathcal A}(T,G,\partial)$$ of a crossed module $$\partial: T\to G$$ to be the crossed module $$\Delta$$ : D(G,T)$$\to Aut(T,G,\partial)$$ where D(G,T) is Whitehead’s (loc. cit.) group of derivations $$G\to T$$, and Aut(T,G,$$\partial)$$ is the group of automorphisms of the crossed module. There is a morphism of crossed modules $$<\eta,\gamma >: (T,G,\partial)\to {\mathcal A}(T,G,\partial)$$, whose image is called the inner actor of the crossed module, and whose kernel is the centre of the crossed module. Section 2 defines actions and semidirect products of crossed modules. Section 3 shows that the above morphism $$<\eta,\gamma >$$ forms part of a crossed square in the sense of D. Guin-Waléry and J.-L. Loday [Algebraic K-theory, Proc. Conf., Lect. Notes Math. 854, 179-216 (1981; Zbl 0461.18007)]. This helps to explain the relation between crossed squares and 2-cat-groups, as in J.-L. Loday [J. Pure Appl. Algebra 24, 179-202 (1982; Zbl 0491.55004)], since a crossed square is also a crossed module internal to the category of crossed modules. This result is also related to work of R. Brown and N. D. Gilbert on automorphism structures for crossed modules, obtained from a different viewpoint [in Proc. Lond. Math. Soc., III. Ser. 59, 51-73 (1989; Zbl 0645.18007)]. Section 4 considers the analogues for crossed modules of various results on completeness of groups, for example results of Burnside and of J. Rose.
{Results of this type have been applied by L. Breen [Théorie de Schreier supérieure (Prepublication Université Paris-Nord 91-3)]. They fit into the notion of ‘Higher order symmetry’, as suggested by Brown and Gilbert (loc. cit.).}
Reviewer: R.Brown (Bangor)

##### MSC:
 20F28 Automorphism groups of groups 18D35 Structured objects in a category (MSC2010) 55Q05 Homotopy groups, general; sets of homotopy classes 55U99 Applied homological algebra and category theory in algebraic topology 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 18G55 Nonabelian homotopical algebra (MSC2010) 19D55 $$K$$-theory and homology; cyclic homology and cohomology
Full Text:
##### References:
 [1] BROWN (R.) . - Higher dimensional group theory, in Low Dimensional Topology , Lond. Math. Soc. Lect. Notes, 48, Cambridge Univ. Press, 1982 , p. 215-238. MR 83k:55014 | Zbl 0484.55007 · Zbl 0484.55007 [2] BROWN (R.) and GILBERT (N.D.) . - Algebraic models of 3-types and automorphism structures for crossed modules , Proc. London Math. Soc. (3), t. 59, 1989 , p. 51-73. MR 90e:18015 | Zbl 0645.18007 · Zbl 0645.18007 [3] BROWN (R.) and HUEBSCHMANN (J.) . - Identities among relations, in Low Dimentional Topology , London Math. Soc. Lect. Notes, t. 48, Cambridge Univ. Press, 1982 , p. 153-202. MR 83h:57008 | Zbl 0485.57001 · Zbl 0485.57001 [4] BROWN (R.) and LODAY (J.L.) . - Van Kampen theorems for diagrams of spaces , Topology, t. 26, 1987 , p. 311-335. MR 88m:55008 | Zbl 0622.55009 · Zbl 0622.55009 [5] BROWN (R.) and LODAY (J.L.) . - Homotopical excision, and Hurewicz theorems , for n-cubes of spaces, Proc. London Math. Soc. (3), t. 54, 1987 , p. 176-192. MR 88c:55017 | Zbl 0584.55012 · Zbl 0584.55012 [6] BURNSIDE (W.) . - The Theory of Groups of Finite Order . - Cambridge Univ. Press, 1897 . JFM 28.0118.03 · JFM 28.0118.03 [7] CONDUCHÉ (D.) . - Modules croisés généralisés de longueur 2 , J. Pure Appl. Algebra, t. 34, 1984 , p. 155-178. MR 86g:20068 | Zbl 0554.20014 · Zbl 0554.20014 [8] DEDECKER (P.) . - Three-dimensional non-abelian cohomology for groups . - Battelle Summer Institute on Homological and Categorical Algebra, Seattle, 1968 . Zbl 0249.18027 · Zbl 0249.18027 [9] GUIN (D.) , WALERY and LODAY (J.L.) . - Obstruction à l’excision en K-théorie algebrique , in Proc. Conf. Algebraic K-Theory Evanston, 1980 . - Lect. Notes in Math., 854, Springer, Berlin, p. 179-216, 1981 . Zbl 0461.18007 · Zbl 0461.18007 [10] HUEBSCHMANN (J.) . - Crossed n-fold extensions of groups and cohomology , Comment. Math. Helvetici, t. 55, 1980 , p. 302-314. MR 82e:20063 | Zbl 0443.18019 · Zbl 0443.18019 [11] HUQ (S.A.) . - Commutator, nilpotency and solvability in categories , Quart. J. Math. Oxford, (2), t. 19, 1968 , p. 363-389. MR 38 #5876 | Zbl 0165.03301 · Zbl 0165.03301 [12] LODAY (J.-L.) . - Cohomologie et groupes de Steinberg relatifs , J. Algebra, t. 54, 1978 , p. 178-202. MR 80b:18013 | Zbl 0391.20040 · Zbl 0391.20040 [13] LODAY (J.-L.) . - Spaces with finitely many non-trivial homotopy groups , J. Pure Appl. Alg., t. 24, 1982 , p. 179-202. MR 83i:55009 | Zbl 0491.55004 · Zbl 0491.55004 [14] LUE (A.S.-T.) . - Crossed homomorphisms of Lie algebras , Proc. Camb. Phil. Soc., t. 62, 1966 , p. 577-581. MR 35 #4267 | Zbl 0147.28302 · Zbl 0147.28302 [15] LUE (A.S.-T.) . - A non-abelian cohomology of associative algebras , Quart. J. Math. Oxford, (2), t. 19, 1968 , p. 159-180. MR 40 #1453 | Zbl 0185.09304 · Zbl 0185.09304 [16] LUE (A.S.-T.) . - The centre of the outer automorphism group of a free group , Bull. London Math. Soc., t. 11, 1979 , p. 6-7. MR 80e:20048 | Zbl 0474.20020 · Zbl 0474.20020 [17] LUE (A.S.-T.) . - Semi-complete crossed modules and holomorphs of groups , Bull. London Math. Soc., t. 11, 1979 , p. 8-16. MR 80g:20049 | Zbl 0416.20030 · Zbl 0416.20030 [18] LUE (A.S.-T.) . - Cohomology of groups relative to a variety , J. Algebra, t. 69, 1981 , p. 155-174. MR 83j:20058 | Zbl 0468.20044 · Zbl 0468.20044 [19] NORRIE (K.J.) . - Crossed modules and analogues of group theorems , Ph.D. thesis, King’s College. - University of London, 1987 . [20] ROSE (J.S.) . - Automorphism groups of groups with trivial centre , Proc. London Math. Soc., (3), t. 31, 1975 , p. 167-193. MR 54 #7619 | Zbl 0315.20021 · Zbl 0315.20021 [21] ROSE (J.S.) . - A Course in Group Theory . - Cambridge Univ. Press, 1978 . MR 58 #16847 | Zbl 0371.20001 · Zbl 0371.20001 [22] SCOTT (W.R.) . - Group Theory . - Prentice Hall, New-Jersey, 1964 . MR 29 #4785 | Zbl 0126.04504 · Zbl 0126.04504 [23] TAYLOR (R.L.) . - Compound group extentions I, continuation of normal homomorphisms , Trans. Amer. Math. Soc., t. 75, 1953 , p. 106-135. MR 15,599d | Zbl 0053.01303 · Zbl 0053.01303 [24] WHITEHEAD (J.H.C.) . - Note on a previous paper entitled On adding relations to homotopy groups , Ann. of. Math., t. 47, 1946 , p. 806-810. MR 8,167a | Zbl 0060.41104 · Zbl 0060.41104 [25] WHITEHEAD (J.H.C.) . - On operators in relative homotopy groups , Ann. of Math., t. 49, 1948 , p. 610-640. MR 10,392c | Zbl 0041.10102 · Zbl 0041.10102 [26] WHITEHEAD (J.H.C.) . - Combinatorial homotopy II , Bull. Amer. Math. Soc., t. 55, 1949 , p. 453-496. MR 11,48c | Zbl 0040.38801 · Zbl 0040.38801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.