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g and h: Group generators

Source: Putnam 1990 B4

July 12, 2013

graph theoryPutnamcollege contests

Problem Statement

Let

G

be a finite group of order

n

generated by

a

and

b

. Prove or disprove: there is a sequence

g_1, g_2, g_3, \cdots, g_{2n}

such that:

(1)

every element of

G

occurs exactly twice, and

(2)

g_{i+1}

equals

g_{i}a

or

g_ib

for

i = 1, 2, \cdots, 2n

. (interpret

g_{2n+1}

as

g_1

.)

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