Book contents
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Applications of the Baker-Hausdorff formula in the theory of finite p-groups
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Summary
Introduction
The Baker–Hausdorff Formula H(x, y) is defined by the equality exey = eH(x,y) for formal power series in non-commuting variables. This formula is an important instrument in the theory of Lie groups giving a local correspondence between a Lie group and its Lie algebra, but we shall not discuss Lie groups in this paper.
The Mal'cev Correspondence makes use of the Baker–Hausdorff Formula to provide a global correspondence (a so-called equivalence of categories) between nilpotent Lie ℚ-algebras and discreet nilpotent ℚ-powered (that is, torsion-free and divisible) groups. Although finite p-groups are neither torsion-free nor divisible, we shall show how the Mal'cev Correspondence can be applied in the theory of finite p-groups.
Under some rather restrictive conditions (like the nilpotency class to be less than p) an analogous correspondence can be established between finite p-groups and Lie rings (the Lazard Correspondence). As G. Higman remarked in his talk at the Congress of Mathematicians in Edinburgh, 1958, these conditions are “… too severe to be used…, …the sort of thing one wants in the conclusion of one's theorem, rather than in the hypothesis.” Nevertheless, we shall also give examples of applications of the Lazard Correspondence. In particular, it can be used for faster reductions to Lie rings and for constructing certain examples, which may be easier for Lie rings.
As a proving ground for applications of the Baker–Hausdorff Formula, we shall discuss results on automorphisms with few fixed points, regular and almost regular ones.
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- Groups St Andrews 1997 in Bath , pp. 460 - 473Publisher: Cambridge University PressPrint publication year: 1999