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vedicguru
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Homework Statement
Prove that if a group G has no subgroup other than G and {e}, then G is cyclic...
Homework Equations
The Attempt at a Solution
we could say that, let a E G - {e} then we construct <a>...
To prove that a group G is cyclic, we must show that there exists an element g in G such that every other element in G can be written as a power of g. This can be done by finding a generator for the group, or by showing that there are no subgroups other than G and the identity element in G.
A cyclic group is a group where every element can be generated by a single element, known as a generator. This means that the group can be written as a sequence of powers of the generator, and that there are no subgroups other than the group itself and the identity element.
No, if a group has any subgroups other than the group itself and the identity element, it cannot be cyclic. This is because a cyclic group must be generated by a single element, and any additional subgroups would contradict this definition.
If a group G has no subgroups other than G and the identity element, then every element in G must be generated by a single element. This means that G is cyclic, as every element can be written as a power of this generator.
No, a non-cyclic group must have at least one subgroup other than itself and the identity element. This is because a non-cyclic group cannot be generated by a single element, so there must be at least one other subgroup in order for every element to be generated.