# Is it necessary to be all Groups representation as a Matrix?

I know that many groups can be represented by matrix.
for example rotation groups can be represented by orthogonal matrices with determinant +1.
But I need to know Are all group representations about matrices?
Can you show me just one group which can be represented by something other?
And also can you show me sample even if representations is not about Vector?

• Every group can be represented as a group of permutations. (Of course, every permutation can be represented as a matrix....) Mar 10, 2014 at 11:48
• I know that the groups can be represented by matrix! but I need to know can groups be represented by non-matrices? Mar 10, 2014 at 11:50
• It is not possible to answer this unless you say exactly what you mean by a group representation. Mar 10, 2014 at 11:54
• And I told you how groups can be represented by non-matrices --- I told you they can be represented as permutations. Mar 10, 2014 at 11:56
• @user1729 I agree. I think the OP really needs to take a bit more time to try to explain what he means with all the terms he uses. Mar 10, 2014 at 12:27

They don't have to be... recall that you only get a matrix when you fix a basis...

For example, let $p=a_3x^2+a_2x+a_1$ be in $V$ the vector space of quadratic polynomials and $\Phi$ the action of $S_3$ on $V$ given by

$$\Phi(p,\pi)=\pi(a_3)x^2+\pi(a_2)x+\pi(a_1).$$

This thing is a representation (unless I am acting on the wrong side... not sure)

Of course this can be realised as a matrix but as a starting point for thinking...

• So you think that the representations might be always matrices?! Mar 10, 2014 at 11:39
• Well any finite dimensional representation can be, once a basis is written down, be considered a matrix. If you want you could work over an infinite dimensional vector space and say that you can't really write these down or whatever... what do you have against matrices anyway? Mar 10, 2014 at 11:40
• even if it is finite group G, we know that there might be many different kind of representations about matrices. but you should think about how to represent this group G by non-matrices. Mar 10, 2014 at 11:46
• What about the one I did above? What you think of that... not a matrix in sight? Mar 10, 2014 at 11:47