[GAP Forum] Work out simple and intuitive examples in GAP to verify the properties of faithful representation.

[Hongyi Zhao]

Hi GAP team,

I noticed the following description here [1]:

```
A representation V of a finite group G over an algebraically closed
field K of characteristic zero is faithful (as a representation) if
and only if every irreducible representation of G occurs as a
subrepresentation of SnV (the n-th symmetric power of the
representation V) for a sufficiently high n. Also, V is faithful (as a
representation) if and only if every irreducible representation of G
occurs as a subrepresentation of

V ⊗ n = V ⊗ V ⊗ ⋯ ⊗ V ⏟ n  times

(the n-th tensor power of the representation V) for a sufficiently high n.
```

The meaning is very abstract and obscure to read and understand, so I
want to know if I can work out simple and intuitive examples in GAP to
verify the properties of faithful representation depicted above.


[1] https://en.wikipedia.org/wiki/Faithful_representation

Regards
-- 
Assoc. Prof. Hongsheng Zhao <hongyi.zhao at gmail.com>
Theory and Simulation of Materials
Hebei Vocational University of Technology and Engineering
No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province