Manoj K. Yadav

On Central Automorphisms Fixing the Center Element-Wise

Let G be a finite p-group of nilpotency class 2. We find necessary and sufficient conditions on G such that each central automorphism of G fixes the center of G element-wise.

Finite groups with many product conjugacy classes

We classify all finite groupsG such that the product of any two non-inverse conjugacy classes ofG is always a conjugacy class ofG. We also classify all finite groupsG for which the product of any twoG-conjugacy classes which are not inverse modulo the center ofG is again a conjugacy class ofG.

On Finite p-Groups Whose Central Automorphisms are all Class Preserving

We obtain certain results on a finite p-group whose central automorphisms are all class preserving. In particular, we prove that if G is a finite p-group whose central automorphisms are all class preserving, then d(G) is even, where d(G) denotes the number of elements in any minimal generating set for G. As an application of these results, we obtain some results regarding finite p-groups whose automorphisms are all class preserving.

On finite p-groups whose automorphisms are all central

An automorphism α of a group G is said to be central if α commutes with every inner automorphism of G. We construct a family of non-special finite p-groups having abelian automorphism groups. These groups provide counterexamples to a conjecture of A. Mahalanobis [Israel J. Math. 165 (2008), 161–187]. We also construct a family of finite p-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in Group Theory, Aracne Editrice, Rome, 2002, pp. 111–127].

Groups St Andrews 2009 in Bath: Class preserving automorphisms of finite p -groups: a survey

In this short survey article, we try to list maximum number of known results on class preserving automorphisms of finite p-groups.

On automorphisms of finite p-groups

Abstract Let G be a finite p-group such that xZ(G) ⊆ xG for all x ∈ G – Z(G), where xG denotes the conjugacy class of x in G. Then |G| divides |Aut(G)|, where Aut(G) is the automorphism group of G.

Estimation of Wheat/Rice Residue Burning Areas in Major Districts of Haryana, India, Using Remote Sensing Data

Haryana has emerged as an important state for Rice & Wheat production in India contributing significantly in the central pool. Mechanized combine harvesting technologies, which have become common in Rice Wheat System (RWS) in India, leave behind large quantities of straw in the field for open burning of residue. Besides causing pollution, the burning kills the useful micro flora of the soil causing soil degradation. There is no field survey (Girdawari) data available with the Government for the areas where stubble burning is taking place. The present paper describes the methodology and results of wheat and rice residue burning areas for three districts of Haryana namely Kaithal, Kurukshetra and Karnal for the year 2010 using complete enumeration approach of multi-date IRS-P6 AWiFS and LISS-III data. In season ground truth was collected using hand held GPS and used to identify area of burnt wheat/rice residues, associated crops and land features. After geo-referencing the satellite images, district images were masked-out and multi-date image data stacks were created. Normalized Difference Vegetation Index (NDVI) of each date was generated and used at the time of classification along with other spectral bands. The non-agricultural classes in the image included: forest, wasteland, water bodies, urban/settlement and permanent vegetation etc. The vector of these non-agriculture classes were extracted from the land use, imported and mask was generated. During the classification non-agriculture area was excluded by using mask of these classes. From this the agricultural area could be separated out. The area was estimated by computing pixels under the classified image mask. In season multi-date AWiFS data along with available single-date LISS-III data between third week of April to last week of May are found to be useful for estimation of wheat residue burning areas estimation. The data between second week of October to last week of November is useful for estimation of rice residue burning areas estimation at district level.

ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP

We construct, for the first time, various types of specific non-special finite p-groups having abelian automorphism group. More specifically, we construct groups G with abelian automorphism group such that γ2(G) < Z(G) < Φ(G), where γ2(G), Z(G) and Φ(G) denote the commutator subgroup, the center and the Frattini subgroup of G respectively. For a finite p-group G with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) Z(G) = Φ(G) is elementary abelian; (ii) γ2(G) = Φ(G) is elementary abelian, where p is an odd prime. We construct examples to show the existence of groups G with elementary abelian automorphism group for which exactly one of the above two conditions holds true.

On the probability distribution associated to commutator word map in finite groups

In this paper, we study the probability distribution associated to the commutator word map. In other words, we study the probability of a given element of a group to be equal to a commutator of two randomly chosen group elements. We compute explicit formulas for calculating this probability for some interesting classes of groups having only two different conjugacy class sizes.

CLASS PRESERVING AUTOMORPHISMS OF UNITRIANGULAR GROUPS

Let UTn (K) be a unitriangular group over a field K. We prove that the group of all class preserving automorphisms of UTn (K) is equal to Inn(UTn (K)) if and only if K is a prime field. Let