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How many ways are there to select a committee of 5 members if at least 1 man and 1 woman must be on the committee?
Find an upper bound for the number of possible states in the game of chess, assuming that draw-by-repetition is enforced if the same position.
The National association of college and university business officers researched the change in university and college endowments from 2007 to 2008.
A certain city has experienced a relatively rapid increase in traffic congestion in recent times. The mayor has decided that it's time.
Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG)
If G is any group, define $:G->G by $(g) = g^-1. Show that G is abelian if an only if $ is a homomorphism.
If K is a normal subgroup of G has index m, show that g^m belongs to K for all g belonging to G.
Elements of Sn can be written in an alternative form, called cycle notation. Starting with 1, we see that s(1) = 3 s(4) = 1, back to the start.
Prove that if G1 and G2 are abelian groups, then the direct product G1 x G2 is abelian.
Let G sub1 and G sub 2 be groups, with subgroups H sub 1 and H sub 2, respectivetly. Show that {(x sub 1, x sub 2).
Let p be a prime number and G a group of order p2 with identity element e. let U ? G and U ? {e} be a subgroup of G. prove that U is cyclic.
Show that if a elements in G where G is a finite group with the identity, e, then there exist n elements in Z+ such that a n =e
Prove that in any abelian group G, the set {g/g²=e} is a subgroup of G. Does this result remain true if G is not abelian? Justify your answer.
Let X be a path-connected space and suppose that every map f: S^1 --> X is homotopically trivial but not necessarily by a homotopy leaving the base.
Let G be any non-Abelian group of order 6. By Cauchy's theorem, G has an element, a, of order 2. Let H = a, and let S be the set of left cosets of H.
Show that the conclusion does not follow if we assume the relation (a.b)i = aibi for just two consecutive integers.
Find order of all elements in s3 , where s3 is the symmetric set of permutations of degree 3.
If G is a group of even order, prove it has an element a ? e satisfying a2 = e.
Let G be a nonempty set closed under an associative product, which in addition satisfies Then G is a group under this product.
Consider the groups Z3 x Z3 and Z9. These are each "integer groups" of order 9. Are they isomorphic or not? Give an explicit reason.
Prove that a group of order p^2, where p is a prime number, must have a normal subgroup of order p.
Suppose H is the only subgroup of order O(H) in the finite group G. Prove that H is a normal subgroup of G.
Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal.
Prove that any element s in Sn which commutes with (1, 2 ...r) is of the form s = (1, 2,....., r)it where i = 0, 1, 2,.......,r, , t is a permutation.
Prove that a group of order 108 must have a normal subgroup of order 9 or 27.