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Abstract Algebra
Let Fk(G) be the k-space of class functions on G, given the inner product
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Assuming that char k = 0, show that M is irreducible iff [χM, χM] = 1, for some kG-module M.
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Assuming that char k = 0, show that M is irreducible iff [χM, χM] = 1, for some kG-module M.
Abstract Algebra
Let Fk(G) be the k-space of class functions on G, given the inner product
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Assuming that char k = 0, show that f = χM for some kG-module M iff [f,χi] is a nonnegative integer for all i.
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Assuming that char k = 0, show that f = χM for some kG-module M iff [f,χi] is a nonnegative integer for all i.
Abstract Algebra
Let Fk(G) be the k-space of class functions on G, given the inner product
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Show that, for any two class functions f, f' ∈ Fk(G), there is a “Plancherel formula”
[f, f'] = (sum over i) [f,χi] [f', χi].
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Show that, for any two class functions f, f' ∈ Fk(G), there is a “Plancherel formula”
[f, f'] = (sum over i) [f,χi] [f', χi].
Abstract Algebra
Let Fk(G) be the k-space of class functions on G, given the inner product
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Show that, for any class function f ∈ Fk(G), there is a “Fourier expansion” f = (sum over i) [f,χi] χi.
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Show that, for any class function f ∈ Fk(G), there is a “Fourier expansion” f = (sum over i) [f,χi] χi.
Abstract Algebra
Show that the First Orthogonality Relation can be generalized to
(sum over g∈G) χ_i(g^−1)χ_j(hg) = δ_ij |G|χ_i(h)/n_i,
where h is any element in G, and n_i = χ_i(1).
(sum over g∈G) χ_i(g^−1)χ_j(hg) = δ_ij |G|χ_i(h)/n_i,
where h is any element in G, and n_i = χ_i(1).
Abstract Algebra
Let G be a finite group such that, for some field k, kG is a finite direct product of k-division algebras. Show that any subgroup H ⊆ G is normal.
Abstract Algebra
Let k be a field, and H be a normal subgroup of a finite group G. Show that rad kG = kG • rad kH iff char k is not divisor of [G : H].
Abstract Algebra
For any field k and for any normal subgroup H of a group G, assume further that [G : H] is finite and prime to char k. Let V be a kG-module and W be a kH-module. Show that W is a semisimple kH-module iff the induced module kG ⊗kH W is a semisimple kG-module.
Abstract Algebra
For any field k and for any normal subgroup H of a group G, assume further that [G : H] is finite and prime to char k. Let V be a kG-module and W be a kH-module. Show that V is a semisimple kG-module iff kHV is a semisimple kH-module.
Abstract Algebra
A new cruise ship line has just launched 3 new ships. The Pacific Paradise, the Caribbean Paradise, & the Mediterranean Paradise. The Caribbean Paradise has 16 more deluxe staterooms than the Pacific Paradise. The Mediterranean Paradise has 40 fewer deluxe staterooms than 4 times the number of deluxe staterooms on the Pacific Paradise. Find the number of deluxe staterooms for each of the ship if the total number of deluxe staterooms for the 3 ships is 836.
