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week02tutsols

# week02tutsols - THE UNIVERSITY OF SYDNEY Math29688...

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THE UNIVERSITY OF SYDNEY Math29688 Algebra (Advanced) Semester 2 Tutorial Solutions Week 2 2008 1. (for general discussion) Explain how the conjugation principle applies in the following situations: (a) using a bus to travel from point A to point B; (b) using an automatic telling machine; (c) taking a cake out of the oven (careful, hot!); (d) the survival of a chameleon. Solution (a) Perform XYX 1 where X is the operation of stepping on the bus and Y is the operation of remaining seated while the bus moves from A to B . (b) Perform XYX 1 where X is the operation of putting your card into the slot and Y is the operation of typing the various keys of the machine. (c) Perform X 1 X 2 YX 1 2 X 1 1 where X 1 is the operation of putting on oven mits, X 2 is the operation of opening the oven door and Y is the operation of taking the cake from the open oven to the kitchen bench. Note that X 1 and X 2 and their inverses can be performed in either order. (d) The chameleon survives by camouflage, so, assuming it starts and finishes in an environment of the same base colour, a portion of its life may be modelled by a sequence of the form X 1 Y 1 X 1 1 X 2 Y 2 X 1 2 ...X k 1 Y k 1 X 1 k 1 X k Y k X 1 k where X i is the operation of changing from a base colour to colour i and Y i is the operation of living whilst in the environment of colour i . A more accurate model would replace each X 1 i X i +1 by the operation of changing directly from colour i to colour i + 1. 2. Express the following permutations of { 1 , 2 , 3 , 4 , 5 , 6 } using cycle notation: α : 1 mapsto→ 3 , 2 mapsto→ 5 , 3 mapsto→ 6 , 4 mapsto→ 2 , 5 mapsto→ 1 , 6 mapsto→ 4 β : 1 mapsto→ 2 , 2 mapsto→ 1 , 3 mapsto→ 3 , 4 mapsto→ 6 , 5 mapsto→ 5 , 6 mapsto→ 4 γ : 1 mapsto→ 4 , 2 mapsto→ 3 , 3 mapsto→ 6 , 4 mapsto→ 5 , 5 mapsto→ 1 , 6 mapsto→ 2 Solution We have α = (136425), β = (12)(46) and γ = (145)(236). 3. Find the composites αβ , αγ and βγ where α , β and γ are the permutations defined in the previous question. Express the inverses of α , β , γ as positive powers of α , β , γ respectively. Solution We have αβ = (134)(25), αγ = (165432), βγ = (1365)(24), α 1 = α 5 , β 1 = β and γ 1 = γ 2 . 4. Let α : 1 mapsto→ 2 mapsto→ 3 mapsto→ 1 and β : 1 mapsto→ 1 , 2 mapsto→ 3 mapsto→ 2. Verify the following equations (where 1 denotes the identity permutation): α 3 = β 2 = 1 , βα = α 2 β, βαβ = α 1 . Put G = ( α,β ) , the group generated by α and β . Deduce that G = { α i β j | 0 i 2 , 0 j 1 } .

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If the letters 1 , 2 , 3 are vertices of an equilateral triangle, interpret elements of G in terms of their geometric effects on the triangle. How many different interpretations of the symbol 1 appear in this exercise? Are you bothered by this?
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week02tutsols - THE UNIVERSITY OF SYDNEY Math29688...

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