By Niki Myrto Mavraki

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We ean leam more as follows. 2. Eaeh rotational symmetry induees a permutation ofthe vertiees, and therefore a permutation of the first four integers. For example, the rotation r illustrated induees the eyclic permutation (234), and s induees (14)(23). Working systematieally through all the other possibilities produees the twelve elements of A 4 • Iftwo rotations u, v induee permutations rx,ß, respeetively then uv clearly induces rxß. Therefore, the eorrespondenee rotational symmetry --+ indueed permutation shows that G is isomorphie to A 4 • 7.

Proof. Mimie the argument of Example (iv). Define q>: H x by q>(x,y) = xy for all x E H, y E K~G K. Then q>«x,y)(x',y'» = q>(xx',yy') = xx'yy' = xyx'y', beeause elements of H eommute with elements of K = q>(x,y)q>(x',y').

5. Let G be a group. Show that the eorrespondenee x+-+x- 1 is an isomorphism from G to G if and only if G is abelian. 6. Prove that 4)pos is not isomorphie to Z. 7. If Gis a group, and if 9 is an element of G, show that the funetion cp: G -+ G defined by cp(x) = gxg- 1 is an isomorphism. Work out this isomorphism when Gis A 4 and 9 is the permutation (123). 8. Call H aproper subgroup ofthe group G if His neither {e} nor all ofG. Find a group whieh is isomorphie to one of its proper subgroups. 9.

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