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Abstract Algebra
Let R = kG where k is any field and G is any group. Let I be the ideal of R generated by ab − ba for all a, b ∈ R. Show that I = (sum over a∈G) (a − 1)kG.
Abstract Algebra
Let R = kG where k is any field and G is any group. Let I be the ideal of R generated by ab − ba for all a, b ∈ R. Show that R/I ∼ k[G/G'] as k-algebras, where G' denotes the commutator subgroup of G.
Abstract Algebra
Compute, the decompositions of QG1, QG2, where G1 is the Klein 4-group, and G2 is the quaternion group of order 8.
Abstract Algebra
Let G = S3. Compute the central idempotents of QG that give this decomposition of QG into its simple components.
Abstract Algebra
Let G = S3. Show that QG ∼ Q × Q ×M2(Q).
Abstract Algebra
Let k be a field whose characteristic is prime to the order of a finite group G. Show that the following two statements are equivalent:
(a) each irreducible kG-module has k-dimension 1;
(b) G is abelian, and k is a splitting field for G.
(a) each irreducible kG-module has k-dimension 1;
(b) G is abelian, and k is a splitting field for G.
Abstract Algebra
Give an example of a pair of finite groups G,G' such that, for some field k, kG ∼ kG' as k-algebras, but G is not isomorphic to G' as groups.
Abstract Algebra
a man walks right,left, back or front with equal probability.find the probability that after 9 steps, he is one step away from the starting point?
Abstract Algebra
Write the sample space for each game. 1.Toss a coin 2.pick a letter from the word random
3.Toss an 8-sided block numbered 1-8 4.pick a marble from a jar containing 3red and 2 white marbles
5.spin the spinner to the right 6.pick a letter from school.
3.Toss an 8-sided block numbered 1-8 4.pick a marble from a jar containing 3red and 2 white marbles
5.spin the spinner to the right 6.pick a letter from school.
Abstract Algebra
what is the decimal answer for the fraction 7 over 20
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#339914 on Dec 2023
#339914 on Dec 2023