I would like to know the idea behind the induction proof in the this question. I added a comment to the answer, but the author has not been active for a year.
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1Use so-called “strong induction”: you assume the result is true for any group of cardinality strictly less than that of $L$, and prove it for $L$; you do not assume it holds if $L$ has cardinality $k$ and then try to prove it for $|L|=k+1$. – Arturo Magidin Apr 06 '19 at 14:29
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@ArturoMagidin good point, thank you. I'll try. – Mohamed Ali Apr 06 '19 at 14:30
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@ArturoMagidin yes, then it is really easy – Mohamed Ali Apr 06 '19 at 14:42
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Which is why the OP just elided it. – Arturo Magidin Apr 06 '19 at 14:49
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1@ArturoMagidin If either you or the OP wanted to submit an answer to this question, I’d happily +1 – Santana Afton Apr 06 '19 at 15:29