SOLUTION: Write True if the statement is correct, Otherwise write FALSE.In each case give a brief explanation. 1. Every abelian group is cyclic. 2. If H and K are subgroups of a group G,

Question 1137638: Write True if the statement is correct, Otherwise write FALSE.In each case give a brief explanation.
1. Every abelian group is cyclic.
2. If H and K are subgroups of a group G, then H intersects K is a group.
3. Every permutation is a cycle.
4. Every function is a permutation if and only if it is one-to-one.
5. Every group, has a unique generator.
Hi The Best Teachers online! Can you help me on this question please? Thank you! God bless!
Answer by ikleyn(52848) (Show Source): You can put this solution on YOUR website!
.

I will give only short answers of the type "TRUE"/"FALSE".

1.  FALSE.

    Counter-examples:

        a)  Abelian groups that are direct sums of other abelian groups, are not cyclic.

        b)  The additive group of all real numbers is not cyclic.

        c)  The multiplicative group of all positive real numbers is not cyclic.



2.  TRUE.

    Make a proof on your own.
    It is easy.


3.  



4.  TRUE.

    If it is the function on discrete finite set, mapping it on itself.



5.  FALSE.

    Counter-examples:

        a)  Abelian groups that are direct sums of other abelian groups, are not cyclic.

        b)  The additive group of all real numbers is not cyclic.

        c)  The multiplicative group of all positive real numbers is not cyclic.

Solved, answered and explained.

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