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 abelian_groups [2021/02/22 20:58]jipsen created abelian_groups [2021/02/22 21:11] Line 1: Line 1: - =====Abelian groups===== - - Abbreviation: **AbGrp** nbsp nbsp nbsp nbsp nbsp [[wp>Abelian group]] - - ====Definition==== - An \emph{abelian group} is a structure $\mathbf{G}=\langle - G,+,-,0\rangle$, where $+$ is an infix binary operation, called the - \emph{group addition}, $-$ is a prefix unary operation, called the - \emph{group negative} and $0$ is a constant (nullary operation), called the \emph{additive identity element}, such that - - $+$ is commutative:  $x+y=y+x$ - - $+$ is associative:  $(x+y)+z=x+(y+z)$ - - $0$ is an additive identity for $+$:  $0+x=x$ - - $-$ gives an additive inverse for $+$:  $-x+x=0$ - - ==Morphisms== - Let $\mathbf{G}$ and $\mathbf{H}$ be abelian groups. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that is a - homomorphism: - $h(x+y)=h(x)+h(y)$ - - Remark: It follows that $h(-x)= -h(x)$ and $h(0)=0$. - - - ====Examples==== - Example 1: $\langle \mathbb{Z}, +, -, 0\rangle$, the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra. - - Example 2: $\mathbb Z_n=\langle \mathbb{Z}/n\mathbb Z, +_n, -_n, 0+n\mathbb Z\rangle$, integers mod $n$. - - Example 3: Any one-generated subgroup of a group. - - - ===Basic results=== - The free abelian group on $n$ generators is $\mathbb Z^n$. - - Classification of finitely generated abelian groups: Every $n$-generated abelian group is isomorphic to a direct product of $\mathbb Z_{p_i^{k_i}}$ for $i=1,\ldots,m$ and $n-m$ copies of $\mathbb Z$, where the $p_i$ are (not necessarily distinct) primes and $m\ge 0$. - - - ====Properties==== - ^[[Classtype]]                       |variety | - ^[[Equational theory]]               |decidable in polynomial time | - ^[[Quasiequational theory]]          |decidable | - ^[[First-order theory]]              |decidable [(Szmielew1949)] | - ^[[Locally finite]]                  |no | - ^[[Residual size]]                   |$\omega$ | - ^[[Congruence distributive]]         |no ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$) | - ^[[Congruence n-permutable]]         |yes, $n=2$, $p(x,y,z)=x-y+z$ | - ^[[Congruence regular]]              |yes, congruences are determined by subalgebras | - ^[[Congruence uniform]]              |yes | - ^[[Congruence types]]                |permutational | - ^[[Congruence extension property]]   |yes, if $K\le H\le G$ then $K\le G$ | - ^[[Definable principal congruences]] |no | - ^[[Equationally def. pr. cong.]]     |no | - ^[[Amalgamation property]]           |yes | - ^[[Strong amalgamation property]]    |yes | - - - ====Finite members==== - ^$n$       | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | - ^# of algs | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 2 |  1 |  1 |  2 |  1 |  1 |  1 |  5 |  1 |  2 |  1 | 1  |  1 |  1 |  1 |  3 |  2 | - ^# of si's | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |  0 |  1 |  0 |  1 |  0 |  0 |  1 |  1 |  0 |  1 | 0  |  0 |  0 |  1 |  0 |  1 | - - see also http://www.research.att.com/projects/OEIS?Anum=A000688 - - - ====Subclasses==== - [[Boolean groups]] - - [[Commutative rings]] - - - ====Superclasses==== - [[Groups]] - - [[Commutative monoids]] - - - ====References==== - - [(Szmielew1949> - W. Szmielew, \emph{Decision problem in group theory}, - Library of the Tenth International Congress of Philosophy, - Amsterdam, August 11--18, 1948, Vol.1, Proceedings of the Congress, - 1949, 763--766 [[http://www.ams.org/mathscinet-getitem?mr=10:500a|MRreview]])]