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Jun 23, 2013 at 21:01 vote accept Max Smith
Jun 23, 2013 at 21:01 comment added Max Smith Thank You, this answers my question. I had similar ideas but I didn't realise that an element like $x_1h$ could be primitive -- bit this is clear to me now. If I am not mistaken, your proof shows that in fact $P\cap H$ generates $H$.
Jun 23, 2013 at 11:30 history answered Derek Holt CC BY-SA 3.0