1. MATH10001 Project 2 Groups part 2 http://www.maths.manchester.ac.uk/undergraduate/ ugstudies/units/2009-10/level1/MATH10001/

2. Subgroups Let G be a group with binary operation . Let H  G with H  . We say that H is a subgroup of G if H is itself a group with respect to . We write H  G.

3. Theorem 1 (Subgroup Test) Let G be a group with binary operation . Let H  G with H  . Then H is a subgroup of G if g  h -1  H for all g, h  H. Example Let G = Z with addition and H = { 4x | x  Z } = {0,  4,  8,…}.

4. Cosets Let H  G and g  G. We define the left coset gH = {g  h | h  H }. Similarly we can define the right coset Hg = {h  g | h  H }. Examples 1.Let G = Z with addition and H = { 4x | x  Z }. 2. Let G = the symmetry group of an equilateral triangle and H = {e, a}.

5. A B C lBlB lAlA lClC e = do nothing a = reflect in line l A b = reflect in line l B c = reflect in line l C r = rotate anticlockwise 120 o s = rotate anticlockwise 240 o A CB C BA B AC C AB B CA

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7. Theorem 2 (Lagrange’s Theorem) Let G be a finite group and H  G. Let n be the number of left cosets of H in G. Then |G| = |H| × n and so |H| divides |G|.