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A non empty subset H of a group G is a subgroup of G if?

Updated: 4/28/2022
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Rose2410

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if and only if H is a group under the group operation of G.

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Q: A non empty subset H of a group G is a subgroup of G if?
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If d divides the order of a group does the group has a subgroup of order d?

The general answer is no. Consider A4={(1),(12)(34),(13)(24),(14)(23),(123),(124),(132),(134),(142),(143),(234),(243)}. The subgroups of A4 are: A4, , , , =, =, =, =, {(1),(12)(34),(13)(24),(14)(23)}, {(1)}. The order of A4 is 12, the order of , and is 2, the order of =, =, = and = is 3, the order of {(1),(12)(34),(13)(24),(14)(23)} is 4, and the order of is 1. Clearly there are no subgroups of order 6, but 6 definitely divides the order of A4. The statement is true for all finite abelian groups, and when d is a power of a prime (i.e., when d=pk for a prime p and a non-negative integer k).


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Related questions

Is A a subset of 2 to the A if a is a non empty set?

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Is a nullset a proper subsets?

The null set is a proper subset of any non-empty set.


What are examples of a proper subset?

The set {1, 3} is a proper subset of {1, 2, 3}.The set {a, b, c, d, e} is a proper subset of the set that contains all the letters in the alphabet.All subsets of a given set are proper subsets, except for the set itself. (Every set is a subset of itself, but not a proper subset.) The empty set is a proper subset of any non-empty set.This sounds like a school question. To answer it, first make up any set you like. Then, as examples of proper subsets, make sets that contain some, but not all, of the members of your original set.


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