Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | Last revision Both sides next revision | ||
group [2015/02/02 19:02] nikolaj |
group [2015/04/16 19:16] nikolaj |
||
---|---|---|---|
Line 11: | Line 11: | ||
----- | ----- | ||
=== Alternative definitions === | === Alternative definitions === | ||
- | Let $\langle G,* \rangle $ be a set $G$ with a binary operation. I'll rewrite the group axioms explicitly in the first order language: | + | == Group axioms explicitly in the first order language == |
+ | Let $\langle G,* \rangle $ be a set $G$ with a binary operation. | ||
1. $\forall (a,b\in G).\ (a*b\in G)$ | 1. $\forall (a,b\in G).\ (a*b\in G)$ | ||
Line 24: | Line 25: | ||
For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "$*$" is defined. | For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "$*$" is defined. | ||
+ | |||
+ | == Sharper definitions == | ||
+ | We could just define left units and left inverses and prove from the group axioms that they are already units and inverses. | ||
----- | ----- | ||
=== Subset of === | === Subset of === | ||
[[Monoid]], [[Loop]] | [[Monoid]], [[Loop]] |