Are applications of group theory known to exist in numerical analysis? One particular aspect I am curious about is whether matrix groups have been successfully used to derive algorithms. Also, are there aspects of numerical analysis that would have been difficult to conceive or derive WITHOUT the help of group theory?
Alain Connes& K mentions some "Butcher group" ( http://arxiv.org/abs/hepth/9904044 ):
See also: On the Hopf Algebraic Structure of Lie Group Integrators H. Z. MuntheKaas, W. M. Wright Hopf algebras of formal diffeomorphisms and numerical integration on manifolds Alexander Lundervold, Hans MuntheKaas Hopf algebras are in some sense "almost groups":) so hopefully this should qualify. In general it seems to me that these series of works by Connes, Kreimer, Broadhurst, Moscovich & K on the Hopf algebras in various fields of math is quite fascinating... 


As far as DFT and FFT is considered as a part of Numerical Analysis, finite abelian groups apply, especially ${\mathbb Z}/2^n{\mathbb Z}$. 


There is a (recent) approach to the problem of fast matrix multiplication that involves representation theory of finite (nonabelian) groups. (Roughly, it reduces/transfers the problem of the existence of fast algorithms to certain question on representations of finite groups.) This was pioneered by Cohn and Umans, see for example Group theoretic algorithms for matrix multiplication by Cohn, Kleinberg, Szegedy, Umans for more details. However, as incidentally recently mentioned on MO by one of the authors these algorithms are not / cannot (yet) be used in practise. 


I recommend P. Winternitz, Group Theory and Numerical Analysis, AMS, 2005. From Google Books:
Also, P.J. Oliver, Applications of Lie Groups to Differential Equations, Springer, 2000.



To solve PDE exactly, you can use the "differential Galois theory" which first was studied by Sofus Lie and more recently by Ovsiannikov and his followers. See, for example Ovsyannikov, L. V. The group analysis purposes. Modern group analysis, V (Johannesburg, 1994). Lie Groups Appl. 1 (1994), no. 1, 193–202. I am sure that similar ideas can work in numerical analysis. 


An example of ideas from Lie groups being used for the analysis of ODE integrators can be found in Reich, S. (1999). Backward error analysis for numerical integrators. SIAM Journal on Numerical Analysis, 36(5), 1549–1570.. 

