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- | =====Abelian groups===== | ||
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- | Abbreviation: | ||
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- | ====Definition==== | ||
- | An \emph{abelian group} is a structure $\mathbf{G}=\langle | ||
- | G, | ||
- | \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 | ||
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- | $+$ is commutative: | ||
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- | $+$ is associative: | ||
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- | $0$ is an additive identity for $+$: $0+x=x$ | ||
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- | $-$ gives an additive inverse for $+$: $-x+x=0$ | ||
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- | ==Morphisms== | ||
- | Let $\mathbf{G}$ and $\mathbf{H}$ be abelian groups. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h: | ||
- | homomorphism: | ||
- | $h(x+y)=h(x)+h(y)$ | ||
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- | Remark: It follows that $h(-x)= -h(x)$ and $h(0)=0$. | ||
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- | ====Examples==== | ||
- | Example 1: $\langle \mathbb{Z}, +, -, 0\rangle$, the integers, with addition, unary subtraction, | ||
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- | Example 2: $\mathbb Z_n=\langle \mathbb{Z}/ | ||
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- | Example 3: Any one-generated subgroup of a group. | ||
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- | ===Basic results=== | ||
- | The free abelian group on $n$ generators is $\mathbb Z^n$. | ||
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- | 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 $i=1, | ||
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- | ====Properties==== | ||
- | ^[[Classtype]] | ||
- | ^[[Equational theory]] | ||
- | ^[[Quasiequational theory]] | ||
- | ^[[First-order theory]] | ||
- | ^[[Locally finite]] | ||
- | ^[[Residual size]] | ||
- | ^[[Congruence distributive]] | ||
- | ^[[Congruence n-permutable]] | ||
- | ^[[Congruence regular]] | ||
- | ^[[Congruence uniform]] | ||
- | ^[[Congruence types]] | ||
- | ^[[Congruence extension property]] | ||
- | ^[[Definable principal congruences]] |no | | ||
- | ^[[Equationally def. pr. cong.]] | ||
- | ^[[Amalgamation property]] | ||
- | ^[[Strong amalgamation property]] | ||
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- | ====Finite members==== | ||
- | ^$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 | | ||
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- | see also http:// | ||
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- | ====Subclasses==== | ||
- | [[Boolean groups]] | ||
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- | [[Commutative rings]] | ||
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- | ====Superclasses==== | ||
- | [[Groups]] | ||
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- | [[Commutative monoids]] | ||
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- | ====References==== | ||
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- | [(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:// | ||