  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YGeneral aims[133X[101X
  
  [33X[0;0YLet  [22XR[122X  be  an  associative  ring,  not necessarily with one. The set of all
  elements  of  [22XR[122X  forms  a monoid with the neutral element [22X0[122X from [22XR[122X under the
  operation [22Xr ⋅ s = r + s + rs[122X defined for all [22Xr[122X and [22Xs[122X of [22XR[122X. This operation is
  called  the  [13Xcircle  multiplication[113X,  and  it  is  also  known  as  the [13Xstar
  multiplication[113X.  The monoid of elements of [22XR[122X under the circle multiplication
  is  called  the  adjoint semigroup of [22XR[122X and is denoted by [22XR^ad[122X. The group of
  all  invertible elements of this monoid is called the adjoint group of [22XR[122X and
  is denoted by [22XR^*[122X.[133X
  
  [33X[0;0YThese  notions  naturally lead to a number of questions about the connection
  between  a  ring and its adjoint group, for example, how the ring properties
  will  determine  properties of the adjoint group; which groups can appear as
  adjoint groups of rings; which rings can have adjoint groups with prescribed
  properties, etc.[133X
  
  [33X[0;0YFor  example,  V. O. Gorlov in [Gor95] gives a full list of finite nilpotent
  algebras [22XR[122X, such that [22XR^2 ne 0[122X and the adjoint group of [22XR[122X is metacyclic (but
  not cyclic).[133X
  
  [33X[0;0YS.  V.  Popovich  and  Ya.  P. Sysak in [PS97] characterize all quasiregular
  algebras   such  that  all  subgroups  of  their  adjoint  group  are  their
  subalgebras.  In  particular,  they  show that all algebras of such type are
  nilpotent with nilpotency index at most three.[133X
  
  [33X[0;0YVarious  connections between properties of a ring and its adjoint group were
  considered by O. D. Artemovych and Yu. B. Ishchuk in [AI97].[133X
  
  [33X[0;0YB.  Amberg  and  L.  S.  Kazarin  in  [AK00]  give  the  description  of all
  nonisomorphic  finite  [22Xp[122X-groups  that can occur as the adjoint group of some
  nilpotent [22Xp[122X-algebra of the dimension at most 5.[133X
  
  [33X[0;0YIn  [AS01]  B.  Amberg  and Ya. P. Sysak give a survey of results on adjoint
  groups  of  radical rings, including such topics as subgroups of the adjoint
  group;  nilpotent  groups  which are isomorphic to the adjoint group of some
  radical  ring;  adjoint groups of finite nilpotent $p$-algebras. The authors
  continued their investigations in further papers [AS02] and [AS04].[133X
  
  [33X[0;0YIn  [KS04]  L. S. Kazarin and P. Soules study associative nilpotent algebras
  over  a  field  of  positive  characteristic whose adjoint group has a small
  number of generators.[133X
  
  [33X[0;0YThe  main objective of the proposed [5XGAP[105X4 package [5XCircle[105X is to extend the [5XGAP[105X
  functionality  for  computations  in  adjoint groups of associative rings to
  make  it  possible  to use the [5XGAP[105X system for the investigation of the above
  described questions.[133X
  
  [33X[0;0Y[5XCircle[105X  provides functionality to construct circle objects that will respect
  the  circle  multiplication  [22Xr  ⋅  s  =  r  +  s + rs[122X, create multiplicative
  structures,  generated  by  such objects, and compute adjoint semigroups and
  adjoint groups of finite rings.[133X
  
  [33X[0;0YAlso  we hope that the package will be useful as an example of extending the
  [5XGAP[105X  system  with new multiplicative objects. Relevant details are explained
  in the next chapter of the manual.[133X
  
  
  [1X1.2 [33X[0;0YInstallation and system requirements[133X[101X
  
  [33X[0;0Y[5XCircle[105X  does  not  use  external  binaries  and,  therefore,  works  without
  restrictions  on  the  type  of  the  operating  system. This version of the
  package  is  designed for [5XGAP[105X4.5 and no compatibility with previous releases
  of [5XGAP[105X4 is guaranteed.[133X
  
  [33X[0;0YTo  use  the  [5XCircle[105X online help it is necessary to install the [5XGAP[105X4 package
  [5XGAPDoc[105X  by  Frank Lübeck and Max Neunhöffer, which is available from the [5XGAP[105X
  site or from [7Xhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[107X.[133X
  
  [33X[0;0Y[5XCircle[105X  is  distributed  in  standard  formats  ([11Xtar.gz[111X,  [11Xtar.bz2[111X,  [11Xzip[111X  and
  [11X-win.zip[111X)  and can be obtained from [7Xhttps://gap-packages.github.io/circle[107X or
  from the [5XGAP[105X homepage. To install the package, unpack its archive in the [11Xpkg[111X
  subdirectory of your [5XGAP[105X installation.[133X
  
