This webpage is the companion page to the article ‘Groups that together with any transformation generate regular semigroups or idempotent generated semigroups’ by J. Araújo, J. D. Mitchell, & Csaba Schneider. On this page you can find GAP code and the output it produces that verify several of the assertions made in the paper.

Follow the steps below to load the functions required to verify the assertions made in the paper:

- download and install GAP 4.7 or higher.
- ensure that you have the latest version of the Semigroups package for GAP (at least version 2.0).
- download the files: companion.g and verify.tst and move them to the GAP directory (the one containing lib, bin, pkg and so on).
- start GAP and load the Semigroups package by typing
`LoadPackage("semigroups");`

. - read the file
companion.g into GAP by typing
`Read("companion.g");`

- all the tests below can be run by typing
`Test("verify.tst");`

in around 1 minute. Alternatively, you can type the commands in the files below individually directly into GAP.

### Groups with the universal transversal property that do not always lead to idempotent generated semigroups [Theorem 1.1 (ii)=>(iii)]

The cyclic group of order 5 [the log file]
.

The dihedral group of order 10 [the log file]
.

The group AGL(1,7) [the log file].

The group PGL(2,7) [the log file].

The group PSL(2,8) [the log file].

The group PGammaL(2,8) [the log file]
.

### Groups with universal transversal property that always lead to idempotent generated semigroups [Theorem 1.1 (iii)=>(i)]

The group AGL(1,5) [the log file].

The group PSL(2,5) [the log file].

The group PGL(2,5) [the log file].

### Groups that satisfy the universal transversal property [Theorem 2.7]

Groups that satisfy the universal transversal property
[the log file] |
[the commands].

### Groups that do not satisfy the universal transversal property [Table 1]

Groups of degree 7 [the log file]

Groups of degree 8 [the log file]

Groups of degree 10 [the log file]

Groups of degree 11 [the log file]

Groups of degree 12 [the log file]

Groups of degree 13 [the log file]

Groups of degree 14 [the log file]

Groups of degree 15 [the log file]

Groups of degree 17 [the log file]

Groups of degree 18 [the log file]

Groups of degree 21 [the log file]

Groups of degree 22 [the log file]

Groups of degree 23 [the log file]