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Group Theory

Continuous Maps, Homomorphisms and Cyclic Groups

Let f: S^n --> S^n be a continuous map. Consider the induced homomorphism f*: H~_n (S^n) --> H~_n (S^n), where H~_n is a reduced homology group. Then from the fact that H~_n (S^n) is an infinite cyclic group, it follows that there is a unique integer d such that f*(u) = du for any u in H~_n (S^n). How exactly does "ther

Normal Subgroup of Order

Modern Algebra Group Theory (LXXV) Normal subgroup of a group The group of order p^2, where p is a prime number Prove that a group of orde

Nilpotent Groups

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. The operation in G is matrix multiplication. (a) Show that G is a group (b) Show that G is a finite p-group (c) Consider the upper central series of G: 1 = Z_0 (G) <= Z_1 (G) <= Z_2 (G) <

Group of order 9

This is the question: Consider small groups. (i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3 (ii) List all groups of order at most 10 (up to isomorphism)

Disjoint cycles and least common multiple

Symmetric groups: G = Sn. (i) Let g1, g2 belong G be two disjoint cycles, and let g = g1g2. Prove that o(g) = lcm { o( g1), o(g2)}, where lcm stands for the least common multiple. (ii) Let g= g1g2 ... gr belong G, where g1,g2, ... gr are disjoint cycles. Prove that o(g) = lcm {o(g1), o(g2), ... o(gr)}. Can you tell me

Path-connected Space : Abelian Group

Let x0 and x1 be points of the path-connected space X. Show that Pi_1(X,x0) is abelian iff for every pair a and b of paths from x0 to x1, we have a'=b', where a'([f])=[a-]*[f]*[a];( a- means the reverse of a.) and [f] belongs to Pi_1(X,x0). a':Pi_1(X,x0)->Pi_1(X,x1).