# 2-Transitive permutation groups by Mazurov V. D. PDF

By Mazurov V. D.

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Without loss of generality 0 is supposed to be a closed subgroup of the group 0(2d, K) of all orthogonal transformaions. Thus, every element U € 0 keeps the standard inner product in R . And besides, the continuity of a implies o{U) = ±1 2 (U £ 0 ) . Main result Recall that two local vector fields £ and r\ are said to be Ck conjugate if there exists a Ck diffeomorphism G such that G^ = r,- Take a ((J^o^-equivariant symplectic form w and a local (0,

For Vi : u = t~2h(z), z=j, where h(z) satisfies the ODE 6ft3ft"" + z2ft4ft" + 6zft4ft' - 66ft2ft'ft'" - 56ft2 (ft")2 + 226ft (ft')2 ft" -aft4ft" - 126 (ft')4 + 6ft5 = 0. • For Vi : z = t, u = x~Ah(z), where ft(^) satisfies the ODE ft" - 726 = 0. • For V2 + V4 : z = xe~l, u = e~4tft(z), where h(z) satisfies the ODE 6ft3ft"" + z2ft4ft" + 9*ft4ft' - 66ft2ft'ft'" - 56ft2 (ft")2 + 226ft (ft')2 ft" -126 (ft')4 + 16ft5 = 0. 42 • For -Va + V4 : z= S T-, u= e4th(z), where h(z) satisfies the ODE bz8h3h"" - 6bz*h2h'h'" - 5bzsh2{h")2 + 22bz8h(h')2h" - 12bz%{h'f +12bzs(h')A + 12bz7h3h'" - 56bz7h2h'h" + 44bz7h{h'f + 36bz6h3h" -56bz6h2(h')2 + 24bz*h3h' + z2hAh" - 7zhAh' + 16/i5 = 0.

Max 4 . The fundamental basis of the technique is that, when a differential equation is invariant under a Lie group of transformations, a reduction transformation exists. The machinery of Lie group theory provides the systematic method to search for these special group-invariant solutions. For PDE's with two independent variables, as it is equation (1), a single group reduction transforms the PDE into ordinary differential equations (ODE's), which are generally easier to solve than the original PDE.

### 2-Transitive permutation groups by Mazurov V. D.

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