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)

Geometry: Names and drawings of polygons of sides 4 through 10

(a) Draw polygons with sides n = 4, 5, 6, 7, 8, 9, 10 for the following three cases. 1- non regular polygon 2- regular polygon 3- a shape that is not a polygon (b) Name the following polygons Number of sides name of polygon ------------------ -------------------- 4 5 6 7 8 9 10

Nilpotent group

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) <=....<= Z_c (G) = G Write explicitly what Z_1 (G), Z_2 (G), and Z_3 (G) are. Show that Z_3 / Z_2 = Z (G/Z_2) Prove that G is nilpotent of class exactly n-1.

Group Theory (LXXV): Prove that a group of order p^2, where p is a prime number, must have a normal subgroup of order p.

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

Group Theory (LXXVI): Find the orbits and cycles of the permutation ( 1 2 3 4 5 )( 8 9 )

Modern Algebra Group Theory (LXXVI) Permutation Groups The Orbits and Cycles of Permutations Find the orbits and cycles of the permutation ( 1 2 3 4 5 )( 8 9 )