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Prerna Chaudhary posted an Question
October 16, 2021 • 06:22 am
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30 points
30 points
- IIT JAM
- Mathematics (MA)
45) let p be a prime. show that if h is a subgroup of a group of order 2p that is not normal, then h has order 2.
45) Let p be a prime. Show that if H is a subgroup of a group of order 2p that is not normal, then H has order 2.
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Let G be a group of order 2p. by Lagrange’s Theorem subgroups of G can only have order 1, 2, p, or 2p The subgroup of order 1 is the trivial group {e}. which is normal The subgroup of order 2p is G itself, which is normal since any product of elements of G belongs to G. if H is a subgroup of order p then index of H in G is 2 .which is normal bcz For a subgroup H of a group of index 2 is normal. therefore order of H is 2 which is not normal
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