# Download PDF by American Mathematical Society: A crash course on Kleinian groups; lectures given at a

By American Mathematical Society

ISBN-10: 0387068406

ISBN-13: 9780387068404

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**Additional info for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco**

**Example text**

Then n' is a permutation representation of G on S' which is equivalent to n. Furthermore, let x E G and i E S, and suppose H(XiX) = HXj, in which case (l)n XiX = j and consequently (i)n x = j. Applying e, we obtain (Hx;)n~ = HX j = H(xjx). Thus n' = nH and the theorem is proved. 7] Transitive and Doubly Transitive Permutation Groups The case H 35 = 1 is of particular importance. 16). We refer to this representation p as the (right) regular representation of G. 2 In the regular representation oJG only the identity elementjixes more than one letter.

Consider the action of HI on S. We can clearly decompose S into the disjoint union of subsets SI' S2' ... , SI with SI = {I}, on each of which HI acts transitively (but not necessarily faithfully). 30) G is doubly transitive if and only if HI acts transitively on S - {I} and hence if and only if t = 2. x). i= I a(x) = tIHII. 2) a(x) = tlHil 1 :( i :( n. 3) I" I i= 1 :ceHt a(x) = I" tlH;! = tnlHII = tlGI. i::::: 1 But on the left we have counted a(x) once for every Hi which contains x. However, x fixes a(x) letters and so is contained in exactly a(x) distinct H;'s.

We also use the notation CH(A) for the subgroup of H left elementwise fixed by the elements of A. Clearly CH(A) is A-invariant. Furthermore, this use of the centralizer notation is consistent with that introduced before. 7. TRANSITIVE AND DOUBLY TRANSITIVE PERMUTATION GROUPS A permutation group G acting on a set S is said to be transitive on S provided for s, s' in S, there exists an x in G such that (s)x = Sf and is said to be doubly transitive on S provided for each set of pairs {SI, S2} and {s~, s;} with Si' s; E S, I ~ i ~ 2, and SI =f.

### A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco by American Mathematical Society

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