example of group G such that o(a) & o(b) is finite but o(ab) is infinite #shorts

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Order of elements O(a) and O(b) is finite but the order of product of element o(ab) is infinite. Order of elements O(a) and O(b) is finite but the order of product of element o(ab) is infinite.
ORDER OF a AND b ARE FINITE BUT ORDER OF ab IS INFINITE ORDER OF a AND b ARE FINITE BUT ORDER OF ab IS INFINITE
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Let G be a group and Phi an automorphism of G. If a in G has order o(a), then o(phi(a)) is o(a) . Let G be a group and Phi an automorphism of G. If a in G has order o(a), then o(phi(a)) is o(a) .
Order of an element || O(a)|O(G) || a^O(G) = e Order of an element || O(a)|O(G) || a^O(G) = e
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