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 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) < ...continues

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 pr ...continues

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 ...continues

Group Theory (LXXVII): Find the orbits and cycles of the permutation ( 1 6 2 5 )( 3 4 )

Modern Algebra Group Theory (LXXVII) Permutation Groups The Orbits and Cycles of Permutations Find the orbits and cycles of the permutation ( 1 6 2 ...continues

Group Theory : Permutation Groups and Disjoint Cycles

Write the given permutation as the product of disjoint cycles ( 1 2 3 4 5 )( 8 9 )

Group Theory (LXXIX): Write the given permutation as the product of disjoint cycles ( 1 6 2 5 )( 3 4 )

Modern Algebra Group Theory (LXXIX) Permutation Groups The Orbits and Cycles of Permutations Write the given permutation as the product of disjoint cycle ...continues

Group Theory (LXXX): Express as the product of disjoint cycles ( 1, 2 , 3 )( 4 ,5 )( 1, 6 , 7, 8 , 9 ) ( 1 ,5 ).

Modern Algebra Group Theory (LXXX) Permutation Groups The Product of Disjoint Cycles ...continues

Group Theory (LXXXI): Express as the product of disjoint cycles ( 1 , 2 )( 1 , 2 , 3 )( 1 , 2 ).

Modern Algebra Group Theory (LXXXI) Permutation Groups The Product of Disjoint Cycles ...continues

Group Theory : Inverses of Cycles and Permutuations

Prove that ( 1, 2, 3, …, n )^(-1) = ( n, n – 1, n – 2, …, 3, 2, 1 )