They illustrate their method with free groups, triangular groups, and finite

cyclic groups, for which they obtain optimal time hypercontractive L2 > Lq inequalities with respect to the Markov process given by the world length and with q an even integer.

For the details of representation rings of Spin groups and

cyclic groups, we refer the reader to standard textbooks on representation theory, e.

There exist finite groups that are semidirect products of cyclic groups and are not determined by their endomorphism semigroups in the class of all groups [10].

On the definability of a semidirect product of cyclic groups by its endomorphism semigroup.

For [GAMMA] being a free product of

cyclic groups of prime order this reconstruction was completed in [3], for free products of arbitrary finite groups in [5].

Among the topics are greatest common divisors, integer multiples and exponents, quotients of polynomial rings, divisibility and factorization in integral domains, subgroups of

cyclic groups, cosets and Lagrange's theorem, the fundamental theorem of finite abelian groups, and check digits.

Popular choices for the group G in discrete logarithm cryptography are the

cyclic groups [[].

Polyrotaxanes are polymers with

cyclic groups threaded onto a polymer chain.

They focus mainly on

cyclic groups, even though some of the results become trivial.

Masuda [22, 23] gave an idea to use a technique of Galois descent to Noether's problem for

cyclic groups [C.

Mathematicians and scientists discuss such matters as rigid abelian groups and the probabilistic method, looking for indecomposable right bounded complexes, kernel modules of cotorsion pairs, upper cardinal bounds for absolute structures, subgroups of totally projective primary abelian groups and direct sums of

cyclic groups, generic endomorphisms of homogeneous structures, special pairs and automorphisms of centerless groups, and some results on the algebraic entropy.

On Noether's problem for

cyclic groups of prime order Akinari HOSHI Communicated by Shigefumi MORI, M.