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--- abelian_groups [2021/02/22 20:59]
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+++ abelian_groups [2021/02/22 21:11]
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-=====Abelian groups=====
-Abbreviation: **AbGrp** [[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,
-, 763--766 [[http://www.ams.org/mathscinet-getitem?mr=10:500a|MRreview]])]

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