2

Part of ICMC 2

Problems(2)

ICMC 2018/19 Round 1, Problem 2

Source: Imperial College Mathematics Competition 2018/19 - Round 1

This question, again, comprises two independent parts.
(i) Show that if

(k+1)

integers are chosen from

\left\{1,2,3,...,2k+1\right\}

, then among the chosen integers there are always two that are coprime.
(ii) Let

A=\left\{1,2,\ldots,n\right\}.

n>11

then there is a bijective map

f: A\to A

with the property that, for every

a\in A

, exactly one of

f(f(f(f(a))))=a

f(f(f(f(f(a)))))=a

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ICMC 2018/19 Round 2, Problem 2

Source: Imperial College Mathematics Competition 2018/19 - Round 2

In the symmetric group

S_n\ (n \geq 3)

G_{a,b}

be the subgroup generated by the 2-cycle

(a\ b)

and the n-cycle

(1\ 2\ \cdots\ n)

. Find the index

\left|S_n : G_{a,b}\right|

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