## Quick Reference

An operation on a set is worth considering only if it has properties likely to lead to interesting and useful results. Certain basic properties recur in different parts of mathematics and, if these are recognized, use can be made of the similarities that exist in the different situations. One such set of basic properties is specified in the definition of a group. The following, then, are all examples of groups: the set of real numbers with addition, the set of non-zero real numbers with multiplication, the set of 2×2 real matrices with matrix addition, the set of vectors in 3-dimensional space with vector addition, the set of all bijective mappings from a set *S* onto itself with composition of mappings, the four numbers 1, *i*, −1,−*i* with multiplication. The definition is as follows: a group is a set *G* closed under an operation ○ such that **1**.*a*, *b* and *c* in *G*, *a* ○ (*b* ○ c)=(*a* ○ *b*) ○ *c*,**2**.*e* in *G* such that *a* ○ *e*=*e* ○ *a*=*a* for all *a* in *G*,**3**.*a* in *G*, there is an inverse element *a*′ in *G* such that *a* ○ *a*′=*a*′ ○ *a*=*e*.The group may be denoted by ⟨*G*, ○⟩, or (*G*, ○), when it is necessary to specify the operation, but it may be called simply the group *G* when the intended operation is clear.

**1**.*a*, *b* and *c* in *G*, *a* ○ (*b* ○ c)=(*a* ○ *b*) ○ *c*,

**2**.*e* in *G* such that *a* ○ *e*=*e* ○ *a*=*a* for all *a* in *G*,

**3**.*a* in *G*, there is an inverse element *a*′ in *G* such that *a* ○ *a*′=*a*′ ○ *a*=*e*.

*Subjects:*
Mathematics.

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