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
The growth of finite subgroups in 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
Abstract
For an infinite p-group with a finite set of generators A, the value of the growth function (of the orders) of finite subgroups f(r) is defined to be the maximal size of a finite subgroup lying in the ball of radius r with respect to A. An equivalence class of the function f does not depend of the choice of a generating system. Growth rates of finite subgroups in p-groups, especially in 2-groups, are under investigation in the paper.
Introduction
As Schmidt knew, any infinite 2-group contains finite subgroups of unbounded order. On the other hand, by the Novikov-Adian theorem, in the free Burnside group of sufficiently large odd exponent n, every finite subgroup has order at most n. Moreover, it can happen that in an infinite p-group, where p is a sufficiently large prime number, all proper subgroups are cyclic of order p.
Thus, the collection of orders of finite subgroups can has exhibit diverse behaviour in various periodic groups, and 2-groups are a special case. For a numerical estimate of the set of finite subgroups, we introduce an asymptotic invariant, namely, the growth of this set. (An analogue could be investigated in other algebraic systems). The case of 2-groups is considered below at greater length.
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- Groups St Andrews 1997 in Bath , pp. 579 - 595Publisher: Cambridge University PressPrint publication year: 1999
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