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The question is presented in the form of photo for its length

As the title states, the task is about proving the number of permutations in group G, which is question 3.

According to what is proven in question 2, there could be two sets ${c, c^2, c^3... c^i-1}$ and ${w, w^2, w^3 ... w^j-1}$ where $i<=n$ and $j<=n$. So I think the first step is about proving $i+j

Thx!

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    Which book does the excerpt come from? – 2017-02-13
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    @pepa.dvorak A Book of Abstract Algebra, 2nd ed. – 2017-02-15
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    by C. C. Pinter? Page? – 2017-02-15
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    @pepa.dvorak yes, page 78-79 – 2017-02-15

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From $(viii)$ one deduces that permutations are characterized by the image of one (arbitrary) element $k_j$. In other words, let there be $n+1$ distinct permutations $p_1,\ldots, p_{n+1}$; since they are distinct we know that $p_i(k_j) \neq p_m(k_j)$ for $i\neq j$ (because if $p_i(k_j) = p_m(k_j)$ then $p_i=p_m$), which however implies, there are at least $n+1$ elements, a contradiction. Hence there are at most $n$ permutations in $G$.

Whilst $(vii)$ implies there are at least $n$ elements in $G$.