We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.
It is well-known that a finite group possesses a universal central extension if and only if it is a perfect group. Similarly, given a prime number p, we show that a finite group possesses a universal ...p′-central extension if and only if the p′-part of its abelianization is trivial. This question arises naturally when working with group representations over a field of characteristic p.
For an arbitrary euclidean field F we introduce a central extension (G(F),Φ) of SL(2,F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order-preserving ...bijections of the convex hull of Q in F. If F=R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2,R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.
FINITELY ANNIHILATED GROUPS CHIODO, MAURICE
Bulletin of the Australian Mathematical Society,
12/2014, Letnik:
90, Številka:
3
Journal Article
Recenzirano
Odprti dostop
In 1976, Wiegold asked if every finitely generated perfect group has weight 1. We introduce a new property of groups, finitely annihilated, and show that this might be a possible approach to ...resolving Wiegold’s problem. For finitely generated groups, we show that in several classes (finite, solvable, free), being finitely annihilated is equivalent to having noncyclic abelianisation. However, we also construct an infinite family of (finitely presented) finitely annihilated groups with cyclic abelianisation. We apply our work to show that the weight of a nonperfect finite group, or a nonperfect finitely generated solvable group, is the same as the weight of its abelianisation. This recovers the known partial results on the Wiegold problem: a finite (or finitely generated solvable) perfect group has weight 1.
On a group, constant functions and left translations by group elements map left cosets into left cosets for every subgroup. We determine classes of groups for which this property of preserving cosets ...characterizes constants and translations, e.g., finite non-abelian groups that are perfect, partitioned, primitive, or generated by elements of prime order p. For certain classes of groups we construct other coset-preserving functions, in particular, power endomorphisms and functions defined in terms of the subgroup lattice.
The second bounded cohomology of a free group of rank greater than 1 is infinite dimensional as a vector space over ℝ
4
. For a group G and its nth commutator subgroup G
(n)
, the quotient G/G
(n)
...is amenable and the homomorphism
induced from the inclusion homomorphism G
(n)
→ G is injective. In this article, we prove that if G
(n)
is free of rank greater than 1 for some finite ordinal n, then G is residually solvable and its second bounded cohomology is infinite dimensional. We prove its converse for a group generated by two elements. As for groups that are not residually solvable, we investigate the dimension of the second bounded cohomology of a perfect group. Also, some results on bounded cohomology of a connected CW complex X by applying a Quillen's plus construction X
+
to kill a perfect normal subgroup of π
1
X are given.
A group G is said to be a "minimal non-FO-group" (an MNFO-group) if all its proper subgroups are FO-groups, but G itself is not. The aim of this article is to study the class of MNFO-groups. The ...structure of MNFO-groups is completely described, both in nonperfect case and perfect case.
Let
G
be a finite group. It is easy to compute the character of
G
corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In ...part this is due to the fact that a class of equivalent representations contains no natural canonical representation.
Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, we know of no general method which has been proposed which is practical for any but small groups.
We shall describe an algorithm for computing an irreducible matrix representation
R
which affords a given character
χ
of a given group
G
. The algorithm uses properties of the structure of
G
which can be computed efficiently by a program such as
GAP, theoretical results from representation theory, theorems from group theory (including the classification of finite simple groups), and linear algebra. All results in this paper have been implemented in the
GAP package
REPSN.
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