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Every subgroup of a cyclic group is cyclic?
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
What is subgroup in mathematics?
In mathematics, a subgroup H of a group G is a subset of G which is also a group with respect to the same group operation * defined on G. H contains the identity element of G, is closed with respect to *, and all elements of H have their inverses in H as well.
When dividing a population into subgroups so that a random sample from each subgroup can be collected?
simple random sampling
Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
How do we prove that a finite group G of order p prime is cyclic using Lagrange?
Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of G with more then 1 element is G itself, therefore G is cyclic. QED.
