The Engel elements in generalized FC-groups

We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.Comment: to appear in "Illinois Journal of Mathematics

An Engel condition for orderable groups

Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular case where v=x the group G is nilpotent (rather than merely locally nilpotent)

Colour gradients of high-redshift Early-Type Galaxies from hydrodynamical monolithic models

We analyze the evolution of colour gradients predicted by the hydrodynamical models of early type galaxies (ETGs) in Pipino et al. (2008), which reproduce fairly well the chemical abundance pattern and the metallicity gradients of local ETGs. We convert the star formation (SF) and metal content into colours by means of stellar population synthetic model and investigate the role of different physical ingredients, as the initial gas distribution and content, and eps_SF, i.e. the normalization of SF rate. From the comparison with high redshift data, a full agreement with optical rest-frame observations at z < 1 is found, for models with low eps_SF, whereas some discrepancies emerge at 1 < z < 2, despite our models reproduce quite well the data scatter at these redshifts. To reconcile the prediction of these high eps_SF systems with the shallower colour gradients observed at lower z we suggest intervention of 1-2 dry mergers. We suggest that future studies should explore the impact of wet galaxy mergings, interactions with environment, dust content and a variation of the Initial Mass Function from the galactic centers to the peripheries.Comment: 13 pages, 7 figures, 1 table, accepted for publication on MNRA

Central dark matter content of early-type galaxies: scaling relations and connections with star formation histories

We examine correlations between masses, sizes and star formation histories for a large sample of low-redshift early-type galaxies, using a simple suite of dynamical and stellar population models. We confirm an anticorrelation between the size and stellar age and go on to survey for trends with the central content of dark matter (DM). An average relation between the central DM density and galaxy size of 〈ρDM〉∝R−2eff provides the first clear indication of cuspy DM haloes in these galaxies – akin to standard Λ cold dark matter haloes that have undergone adiabatic contraction. The DM density scales with galaxy mass as expected, deviating from suggestions of a universal halo profile for dwarf and late-type galaxies. We introduce a new fundamental constraint on galaxy formation by finding that the central DM fraction decreases with stellar age. This result is only partially explained by the size–age dependencies, and the residual trend is in the opposite direction to basic DM halo expectations. Therefore, we suggest that there may be a connection between age and halo contraction and that galaxies forming earlier had stronger baryonic feedback, which expanded their haloes, or lumpier baryonic accretion, which avoided halo contraction. An alternative explanation is a lighter initial mass function for older stellar populations

A restriction on centralizers in finite groups

For a given m>=1, we consider the finite non-abelian groups G for which |C_G(g):|<=m for every g in G\Z(G). We show that the order of G can be bounded in terms of m and the largest prime divisor of the order of G. Our approach relies on dealing first with the case where G is a non-abelian finite p-group. In that situation, if we take m=p^k to be a power of p, we show that |G|<=p^{2k+2} with the only exception of Q_8. This bound is best possible, and implies that the order of G can be bounded by a function of m alone in the case of nilpotent groups