[GAP Forum] a very elementary problem

[Thomas Breuer]

Dear GAP Forum,

Jon Barkhurst wrote

> I'm a new user to gap. ( I got my MS 30 years ago and am getting up to 
> speed).  I'm trying to show that the units mod 20 are isomorphic to C2 X C4. 
>   I can't get my script to work.  would someone give me a script, ( 
> collection of commands) that works?   Many thanks.  Jon Barkhurst

The following works.

    gap> R:= Integers mod 20;;
    gap> u:= Units( R );;
    gap> IsAbelian( u );
    true
    gap> AbelianInvariants( u );
    [ 2, 4 ]

Here are two alternatives.

    gap> StructureDescription( u );
    "C4 x C2"
    gap> IdGroup( u );
    [ 8, 2 ]
    gap> IdGroup( DirectProduct( CyclicGroup(2), CyclicGroup(4) ) );
    [ 8, 2 ]

The function `StructureDescription' returns a string that gives information
about the structure of the group.
For abelian groups, this is enough to determine the isomorphism type.

The function `IdGroup' returns a list that identifies the isomorphism type
of the given groups, that is, the result for two groups is equal if and only if
they are isomorphic.
`IdGroup' works only for groups that are small or that have a special structure.

More information about these GAP functions can be found in the Reference Manual.

All the best,
Thomas