Abstract Algebra Answers

If a finite group G has at most three irreducible complex representations, show that G ∼ {1},Z2,Z3 or S3.

Show that any quasi-Frobenius ring is left (and right) Kasch.

8y^4-4y^2=0

course: abstract math
book: mathematical proofs: a transition to advanced mathematics 3rd edition

Question: For the sets A={1,2,...,10} and B={2,4,6,9,12,25}, consider the statements P: A ⊆ B. Q:|A-B|=6.

Determine which of the following statements are true.
(a) P∨Q
(b) P∨(~Q)
(c) P∧Q
(d) (~P)∧Q
(e) (~P)∨(~Q)

a^4+2a^3+a^2=0

Xuxian and Rachel had the same number of marbles. After Rachel gave away 10 marbles and Xuxian gave away 22 marbles, Rachel had 3 times as many marble as Xuxian. How many marbles did each of them have at first

show that for every a,b in Q if a.b=a+b+ab, then (Q,.) is a group.

Show that for any prime p, a group G of order p2 must be abelian.

Show that, for any two groups G,H, there exists a (nonzero) ring R such that RG ∼ RH as rings.

For finite abelian groups G and H, show that RG ∼ RH as R-algebras iff |G| = |H| and |G/G2| = |H/H2|.