The photochemical synthesis of 4-acyl-1,2-dimethyl-3H-phenothiazin-3-ones

4-Acyl-1,2-dimethyl-3H-phenothiazin-3-ones were prepared by the photochemical reaction of 1,2-dimethyl-3H-phenothiazin-3-one with aldehydes. The structures of the newly prepared compounds were determined by elemental analysis, spectroscopic methods (ir, nmr and ms) and comparison with a sample prepared by an alternate route.     

Photochemical reactions of 6-alkylamino- and 6-alkylanilino-5H-benzo[a]phenothiazin-5-ones

Photochemical reactions of 6-alkylamino- and 6-alkylanilino-5H-benzo[a]phenothiazin-5-ones have been investigated. 6-Ethylamino- and 6-isopropylamino-5H-benzo[a]phenothiazin-5-ones and 6-methylanilino- and 6-ethylanilino-5H-benzo[a]phenothiazin-5-ones gave 6-amino-5H-benzo[a]phenothiazin-5-one and 6-anilino-5H-benzo[a]phenothiazin-5-one, respectively, on irradiation.     

Trivial source modules in blocks with cyclic defect groups

We shall present a method to get trivial source modules easily just by looking at values of ordinary characters at non-identity p-elements in finite groups instead of doing huge calculation. The method is only for a case where defect groups are cyclic. Nevertheless, it works well at least when we want to prove Broué’s abelian defect group conjecture for blocks which have elementary abelian defect groups of order p ².     

The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group

We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k _G ) of the trivial kG-module k _G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined.     

The photochemical reactions of 5H-benzo[a]phenoxazin-5-one with alkylthiols and thiophenol

6-Alkylthio- and 6-phenylthio-5H-benzo[a]phenoxazin-5-ones have been synthesized by the photochemical reaction of 5H-benzo[a]phenoxazin-5-one with alkylthiols and thiophenol.     

The photochemical reactions of 5H-benzo[a]phenothiazin-5-one with alkylthiols and thiophenol

6-Alkylthio- and 6-phenylthio-5H-benzo[a]phenothiazin-5-ones were prepared by the photochemical reaction of 5H-benzo[a]phenothiazin-5-one with alkylthiols and thiophenol. The structures of newly prepared compounds were determined by elemental analysis, spectroscopic methods (ir, nmr and ms) and comparison with a sample prepared by an alternate route.     

Blocks with nonabelian defect groups which have cyclic subgroups of index p

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M _n+1(p) of order p ⁿ⁺¹ and exponent p ⁿ for n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M _n+1(p) is normal in G. The proof is independent of the classification of finite simple groups.