Flower Art by DarleneFloral Crafts Direct From the Artist

Abstract / Modern Algebra: Prove that a group with at least 2 elements, but not trivial proper sungroups should?

Abstract / Modern Algebra: Prove that a group with at least 2 elements, but not trivial proper sungroups must be finite and of order prime.

If G has at least two elements, then choose one, for example g, different from and identity, and consider the group generated by g, and g ≠ e, this is a nontrivial subgroup of G, then we must have G = . (I) If G is infinite, then so is , But is an infinite proper subgroup , Thus also of G, which implies = = G, and this is impossible, so that G is finite. (II) by the Lagrange theorem, | | Must divide | G |, but, if | |> 1, then | | = | G |, hence | G | has no divisors, and is greater than 1, so it must be a prime number.

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