$#C+1 = 60 : ~/encyclopedia/old_files/data/A110/A.1100020 Abelian difference set
...n (cyclic, non-Abelian), the difference set is called Abelian (cyclic, non-Abelian). Two difference sets $ D _ {1} $
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...ecreased. The Thue–Siegel–Roth theorem is a strengthening of the Liouville theorem (see [[Liouville number|Liouville number]]). Liouville's result has been su
...was obtained by Roth. There is a $p$-adic analogue of the Thue–Siegel–Roth theorem. The results listed above are proved by non-effective methods (see [[Diopha
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The [[characteristic subgroup]] of a [[P-group|$p$-group]] generated by all Abelian subgroups of maximal order. Introduced by J.G. Thompson [[#References|[1]]]
...D valign="top">[1]</TD> <TD valign="top"> J.G. Thompson, "A replacement theorem for $p$-groups and a conjecture" ''J. Algebra'' , '''13''' (1969) pp. 14
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...Abelian group. Examples of commutative group schemes are [[Abelian scheme|Abelian schemes]] and [[Algebraic torus|algebraic tori]]. A generalization of algeb
$$ where $M$ is an Abelian group and ${\mathcal O}_S(M)$ is its group algebra with coefficients in the
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...ory of Abelian groups and the category that is dual to that of topological Abelian groups; the category of Boolean algebras is equivalent to the category that
1 KB (231 words) - 07:37, 28 November 2017
$#C+1 = 92 : ~/encyclopedia/old_files/data/A010/A.0100240 Abelian integral,
of the [[Abelian differential|Abelian differential]] $ \omega = R (z, w) dz $
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...ferences|[a2]]], M. Hall proved the following generalization of Frobenius' theorem: If $G$ is a finite group of order $g$ and $C$ is a [[conjugacy class]] of
...ns. Thus, the conjecture holds in Abelian groups (cf. also [[Abelian group|Abelian group]]). It is also easy to see that it suffices to show that $G$ contains
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On the other hand, there exists an amalgam of four Abelian groups that is not imbeddable in a group. The principal problem concerning
...algam of five Abelian groups which is imbeddable in a group, but not in an Abelian group. Another problem that has been studied is the imbeddability of an ama
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...uations by means of radicals. It is customary to write the operation in an Abelian group in additive notation, i.e. to use the plus sign ($+$) for that operat
...up]]s $\mathbf Z_{p^\infty}$), where $p$ is an arbitrary prime number, are Abelian (cf. [[Group-of-type-p^infinity|Group of type $p^\infty$]]).
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...clopedia/old_files/data/G110/G.1100050 Gamma\AAhinvariant in the theory of Abelian groups,
...ubgroup of strictly smaller cardinality is a [[free Abelian group]]). By a theorem of S. Shelah (see [[#References|[a7]]]), such a group is free if it is of [
6 KB (845 words) - 19:41, 5 June 2020
$#C+1 = 90 : ~/encyclopedia/old_files/data/A010/A.0100200 Abelian category
...me of the characteristic properties of the category of all Abelian groups. Abelian categories were introduced as the basis for an abstract construction of hom
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...blem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups $ A $
is free (see [[Free Abelian group|Free Abelian group]]). This condition has been proved to be necessary if $ A $
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$#C+1 = 42 : ~/encyclopedia/old_files/data/F041/F.0401790 Frobenius theorem
...ors of zero; it was proved by G. Frobenius [[#References|[1]]]. Frobenius' theorem asserts that:
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$#C+1 = 145 : ~/encyclopedia/old_files/data/A010/A.0100210 Abelian differential
...g on the nature of their singular points, one distinguishes three kinds of Abelian differentials: I, II and III, with proper inclusions $ I \subset II \sub
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Then the category of sheaves of Abelian groups on $ X _ {et} $
is an Abelian category with a sufficient collection of injective objects. The functor $
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of an [[Abelian category|Abelian category]] $ \mathfrak A $
is Abelian.
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extensions with an Abelian Galois group (Abelian extensions) is a part
The fundamental result on Galois groups is the following theorem,
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...are isomorphic over a finite extension of $K$. One of the marvels of this theorem is the fact that the construction of the period $q$ starting from $E$, and
...], and it was used in the theory of compactifications of moduli schemes of Abelian varieties.
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$#C+1 = 99 : ~/encyclopedia/old_files/data/A010/A.0100220 Abelian function
is called an Abelian function if there exist $ 2p $
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coincides with the set of one-dimensional non-Abelian cohomology $ H ^{1} ( S _{T} ,\ \Gamma ) $.
group is trivial (Lang's theorem). This theorem also holds if $ k $
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