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The symplectic group Sp2n (R) is composed of all 2n×2n symplectic matrices with real entries. groups 19 These groups GLn (R), SLn (R), On (R), SOn (R), Sp2n (R), Un and SUn are, for n > 1 at least, all infinite and nonabelian. But they also have a continuous structure inherited from R, which means that we can regard them as being locally similar to Rm for some value of m. Such an object is called a manifold, and a group which has a compatible manifold structure in this way is known as a Lie group, after the Norwegian mathematician Marius Sophus Lie (1842–1899).

This gives us another trivial cycle (4). And now we’ve dealt with all of the numbers in {1, 2, 3, 4, 5}, so that’s the end, and our permutation can therefore be written as (1)(2 3 5)(4). Except that we’re really only interested in the nontrivial bits of the permutation, which in this case is the 3–cycle (2 3 5); the 1–cycles (1) and (4) don’t tell us anything nontrivial about the permutation, so we discard them, leaving us with σ = (2 3 5). This compact notation tells us that σ causes the numbers 2, 3 and 5 to cycle amongst themselves, and doesn’t do anything to the remaining numbers 1 and 4.

N} is determined completely by the way it maps the numbers amongst themselves. There are n The Old Vicarage, Grantchester groups choices for where 1 maps to, then n−1 choices for where 2 goes (since we can map 2 to any of the remaining numbers except for the one we mapped 1 to), n−2 choices for where 3 maps to, and so on. This gives us n(n−1)(n−2) . . 1 = n! possible permutations of n objects, and so |Sn | = n!. It so happens that S2 ∼ = Z2 and S3 ∼ = D3 . The symmetric group S4 is isomorphic to the symmetry group of the tetrahedron.

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