Abstract Algebra using crongruences for dates
How many times in 1900 did the 1st day of the month fall on a Tuesday? Hint: The year 1900 was not a leap year.
Let R = C([0, 1]) be the ring of continuous real-valued functions on the interval [0, 1], with the usual definitions of sum and product of functions from calculus. Show that f in R is a zero divisor if and only if f is not identically zero and { x | f(x) = 0 } contains an open interval. What are the idempotents of this ring? W ...continues
Let G be the direct sum of a countably infinite number of copies of Z. Find an element of End_Z(G) which has a left inverse, but is not a unit. Please explain in detail. Think of elements of End_Z(G) as infinite matrices with integer entries. Definition: Let G be an abelian group and let End_Z(G) be the set of all gro ...continues
Simplifying rational expressions and least common multiple
Please see the attached document about simplifying rational expressions and LCM
let f: A->A, whereaA is nonempty. Prove that f has a left inverse if and only if f^-1(f(S))=S for every subset S of A
Assume that G is a finite group, and let H be a nonempty subset of G; prove that H is closed iff H is a subgroup of G
let G be a finite group with K is a normal subgroup of G. If (l K l, [G:K]) =1, prove that K is unique subgroup of G having order l K l. I am having trouble with this proof and I need it written in complete proof form and I don't really know how to do it.
Let G be a group, let a, b be elements in G and let m and n be (not necessarily positive) integers. Prove that: i) (a^n)^m= (a^mn) ii) if a and b commute, then (ab)^n = a^n times b^n iii) a^m times a^n = (a)^(m+n)
i) Let A, B, and C be groups, let alpha, beta, and gamma be homomorphisms with gamma times alpha = beta alpha gamma beta A--------->B----------->C<---------A If alpha is surjective, prove that ker(gamma)= alpha((ker(beta)). ii) Prove that if K is a subgroup of a group G, and if every left coset of ...continues
Modern Algebra : Disjoint Proof
Let H be a subgroup of the group G. Prove that if two right cosets Ha and Hb are not disjoint, then Ha = Hb.
