...\mathbf{Q}$ such that the [[Galois group]] $\mathrm{Gal}(K/\mathbf{Q})$ is Abelian. Examples include: the quadratic number fields $\mathbf{Q}(\sqrt{d})$ and
...such that $K$ is contained in $\mathbf{Q}(\zeta_n)$, cf. [[Conductor of an Abelian extension]].
813 bytes (123 words) - 20:47, 23 November 2023
..., then the validity of P in some particular category (e.g. the category of
Abelian groups or the category of sets) implies its validity in all categories of t
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Freyd, "
Abelian categories: An introduction to the theory of functors" , Harper & Row
1 KB (173 words) - 17:22, 7 February 2011
An [[Abelian group|Abelian group]] is cotorsion-free if it does not contain any non-zero [[Cotorsion g
...cyclic group]] of prime order (thus, it is torsion-free). Equivalently, an Abelian group $ A $
4 KB (561 words) - 17:31, 5 June 2020
...bgroup series]]). It also possesses a [[Normal series|normal series]] with Abelian quotient groups (such series are called solvable). The length of the shorte
...er. These groups are characterized by the following converse to Lagrange's theorem: For any factorization $n=n_1n_2$ of the order $n$ of a group into two rela
3 KB (443 words) - 18:25, 26 October 2014
...idèle class group $C_K$ (cf. [[Class field theory]]). The conductor of an Abelian extension is the greatest common divisor of all positive divisors $n$ such
...extension is given by the theorem that the conductor $\mathfrak{f}$ of an Abelian extension $L/K$ of number fields is equal to $\prod_{\mathfrak{p}} \mathfra
2 KB (440 words) - 19:42, 7 March 2018
An [[Abelian group|Abelian group]] $ C $
for all [[Torsion-free group|torsion-free]] Abelian groups $ G $,
4 KB (674 words) - 17:31, 5 June 2020
has an elementary Abelian subgroup of order $ p ^ {n} $(
cf. [[Abelian group|Abelian group]]). A $ p $-
4 KB (568 words) - 14:10, 31 December 2020
[[Brauer first main theorem|Brauer first main theorem]].
...gers with $\def\a{\alpha}\nu(np^\a)=\a$ whenever $n$ is prime to $p$. By a theorem of Brauer, $\nu(\chi(1)\ge \nu(|G:D|)$. The height of $\chi$ is defined to
2 KB (352 words) - 14:21, 13 April 2012
...categories similar to an [[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let $
group (in particular, for groups, Abelian groups, vector spaces, and rings) one can give an "intrinsic" characteriz
4 KB (680 words) - 19:35, 5 June 2020
For finite-dimensional Lie algebras over a field of characteristic 0, Lévy's theorem holds: If $ S $
...ions, the Abelian ones have been studied most, i.e. the extensions with an Abelian kernel $ A $.
3 KB (416 words) - 12:53, 19 March 2023
$#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} $
5 KB (803 words) - 06:20, 26 March 2023
...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
4 KB (557 words) - 18:10, 23 November 2014
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
724 bytes (97 words) - 14:51, 8 April 2023
...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
4 KB (629 words) - 20:08, 15 December 2020
...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 $
10 KB (1,594 words) - 06:20, 17 April 2024
...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
4 KB (650 words) - 20:59, 29 November 2014
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
5 KB (883 words) - 16:10, 1 April 2020
...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 $p^\infty$]]).
11 KB (1,810 words) - 22:12, 29 August 2015
...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