A block cipher is a bijective map from the set of possible plaintexts to the set of ciphertexts, which are the same size and might as well be considered the same thing: $\theta: S\to S$. In this there must exist a fixed point m $\in S$ such that $\theta(m) = m$ (alternatively you could consider encryption to be a binary operator acting on the set $S$ then you could write encryption under key $k$ as $m \odot k = m$, with $m,k \in S$. Unfortunately this cannot form a group since there is no unit $e$ that makes $e \odot k = e$ for all $k \in S$).
It is clear that when the block size is equal to the key size a single fixed point will exist when you hold the plaintext message constant and permute the key. The key allows you to select one of all the possible possible unique mappings, of which one must surely be a fixed point (am I right about my use of the word unique here?).
What I don't understand is how a key that is longer than a block size provides any extra security. From what I understand this would suggest the existence of many fixed points, or equivalently many different keys that will decrypt a ciphertext.
Note: I would also be interested to see articles that apply algebra to the study of block ciphers. Specifically constructing groups to aid analysis that also consider the fact that a block cipher is a composition of several round functions.