SOLUTION: Write True if the statement is correct, Otherwise write FALSE.In each case give a brief explanation. 1.The empty set can be considered a group. 2. Every group is a binary alge

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Write True if the statement is correct, Otherwise write FALSE.In each case give a brief explanation. 1.The empty set can be considered a group. 2. Every group is a binary alge      Log On


   

Question 1137637: Write True if the statement is correct, Otherwise write FALSE.In each case give a brief explanation.
1.The empty set can be considered a group.
2. Every group is a binary algebraic structure.
3. Every group has exactly two improper subgroups.
4. Z4 is a cyclic group.
5. In every cyclic group, every element is a generator.
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Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will answer questions 4 and 5 only. 


4.  TRUE.

    Its generator is the element "1".



5.  FALSE.


    Counter-example.

    In Z4, the element "2" is not its generator.


Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

1.The empty set can be considered a group. FALSE
2. Every group is a binary algebraic structure. TRUE
3. Every group has exactly two improper subgroups. FALSE
4. Z4 is a cyclic group.TRUE
5. In every cyclic group, every element is a generator. FALSE