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A generalization of the concept of a [[Group|group]] to the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660002.png" />-ary operation. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660004.png" />-group is a [[Universal algebra|universal algebra]] with one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660005.png" />-ary associative operation that is uniquely invertible at each place (cf. [[Algebraic operation|Algebraic operation]]). The theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660006.png" />-groups for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660007.png" /> substantially differs from the theory of groups (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660008.png" />-groups). Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n0660009.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600010.png" />-group has no analogue of the unit element.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600011.png" /> be a group with multiplication operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600012.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600013.png" /> be an arbitrary integer. Then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600014.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600015.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600016.png" /> can be defined as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600017.png" /></td> </tr></table>

The resulting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600018.png" />-group is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600020.png" />-group determined by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600021.png" />. Necessary and sufficient conditions for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600022.png" />-group to be of this form are known [[#References|[1]]]. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600023.png" />-group is imbeddable in such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600024.png" />-group (Post's theorem).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>



====Comments====
The usual notion of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600027.png" />-group (i.e., a group of order a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600028.png" />) is not to be mixed up with that of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600029.png" />-group in the above sense.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Balci,   "Zur Theorie der topologischen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600030.png" />-Gruppen" , Minerva , Munich  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.A. Rusakov,   "The subgroup structure of Dedekind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600031.png" />-ary groups" , ''Finite groups (Proc. Gomel. Sem.)'' , Minsk  (1978)  pp. 81–104  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.A. Rusakov,   "On the theory of nilpotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066000/n06600032.png" />-ary groups" , ''Finite groups (Proc. Gomel. Sem.)'' , Minsk  (1978)  pp. 104–130  (In Russian)</TD></TR></table>