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- Prove that any two 2-dimensional lattices are isomorphic as groups.

Under what conditions are these groups conjugate subgroups of*I*(**R**^{2}) ? - A plane lattice is called
*isosceles*if it is of the form {*a**e*_{1}+*b**e*_{2}|*a*,*b*∈**Z**} where*e*_{1}and*e*_{2}are vectors of the same length.

Prove that a lattice can have a reflection or glide as a symmetry if and only if it is either rectangular or isosceles. - Let
*L*be the lattice {*a**e*_{1}+*b**e*_{2}|*a*,*b*∈**Z**} in**R**^{2}. If*g*∈*I*(**R**^{2}) is a symmetry which maps*L*to itself, prove that the matrix*M*_{g}of*g*with respect to the basis {*e*_{1},*e*_{2}} has integer entries.

Is*M*_{g}an orthogonal matrix ?

Use the fact that the*trace*(the sum of the diagonal entries) of a matrix is left fixed by any change of basis to prove the*crystallographic restriction*(Rotations which map a lattice to itself can only have orders 1, 2, 3, 4 or 6). - Let
*U*,*V*be subgroups of a group*G*with*UV*=*G*. That is, every element*g*∈*G*can be written as*g*=*uv*with*u*∈*U*and*v*∈*V*. Prove that if*U*∩*V*= {1} then the terms in this product are uniquely determined.If a group

*G*is isomorphic to a direct product*U*_{1}×*V*_{1}prove that there are*normal*subgroups*U*,*V*of*G*with*U*≅*U*_{1},*V*≅*V*_{1},*UV*=*G*and*U*∩*V*= {1}.Prove conversely that if there are

*normal*subgroups*U*,*V*of*G*with*UV*=*G*and*U*∩*V*= {1} then*G*is isomorphic to a direct product*U*×*V*. - Let
*U*be the subgroup < (123) > and let*V*be the subgroup < (12) > of*G*=*S*_{3}. Prove that*UV*=*G*and*U*∩*V*= {1} but that*G*is not a direct product of*U*and*V*.Prove that the group

*S*_{5}is not a direct product of*A*_{5}and any subgroup of order 2.Let

*T*be the subgroup of translations in*I*(**R**^{n}) and let*O*(*n*) be the subgroup of linear symmetries. Prove that*I*(**R**^{n}) is not a direct product of*T*and*O*(*n*) for*n*≥ 1.

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