Below are one student's answers to the "Concepts" questions in the exercises.  I've put my responses below each of the student's answers.

 

 

Section 4 - #25.

 

  1. There is only one identity element.

 

Right!  (We usually write "exactly one" or "a unique" instead of "only one.")

 

  1. Any 2 groups of 3 elements may or may not be isomorphic. Ex: {234} & {569}

 

Actually, it is true that any two groups of order three are isomorphic.  You might try proving that---it would be a good exercise.

 

  1. In a group, each linear equation has a solution.

 

I don't know what "linear equation" means for a group.  I think what they're fishing for is the statement of Thm. 4.16.

 

  1. Every finite group with less than 3 elements is abelian.

 

What you've written is true; what's written in the book is true, too.  You might try proving it---it would be a good exercise.

 

  1. An equation of the form  a  x   b = c may not always have a unique solution in a group.

 

Actually, it does always have a unique solution.  Namely, x = a^(-1) * c * b^(-1).

 

  1. The empty set can be considered a group. 

 

Nope.  A group must contain an identity element.

 

  1. Every group may or may not be a binary algebraic structure. (it can have one or 3 or more elements).

 

A binary algebraic structure is just a set with a binary operation.  (p. 29)  So, yes, every group is a binary algebraic structure.

 

 

 

Section 5 - # 39.

 

  1. The associative law holds in every group. (if not it is not a group).

 

Right.

 

  1. There are no groups in which the cancellation law fails.

 

Right.

 

  1. Every group is a subgroup of itself and it’s called the improper subgroup.

 

Right.

 

  1. Every group has only one improper subgroup, itself.

 

Right.

 

  1. In every cyclic group, every element may not be a generator.

 

A better way to phrase this is ". . . it is not necessarily true that every element is a generator."  (In fact, there is only one cyclic group G for which it is true that every element is a generator of G.  Can you figure out what G is?)

 

  1. A cyclic group may have many generators.

 

Right.

 

  1. Every set of numbers under addition, may not be a group under multiplication. Ex: (Z ,+) is a group; (Z , )  is not a group.

 

A better way to phrase this is "Not every set of numbers that is a group under addition is also a group under multiplication."  Your example is a good one.  (In fact, if G is a set of numbers that is a group under addition and G has more than one element, then G cannot be a group under multiplication.  Why not?  Hint: Find three numbers a, b, and c for which the cancellation law fails.)

 

  1.  A subgroup may be defined as a subset of a group.

 

I'm not sure what they're getting at with this one.  Maybe they just want you to write the definition of a subgroup.

 

  1. Z4 is a cyclic group.

 

Rught.

 

  1. Every subset of every group may not be a subgroup under the induced operation.

 

A better way to phrase this is "Not every subset of every group is a subgroup under the induced operation."  Another true statement would be " Every subgroup of every group is a subgroup under the induced operation."

 

Section 6 - # 32.

 

  1. Every cyclic group is abelian.

 

Right.

 

  1. Not every abelian group is cyclic.

 

Right.

 

  1. Q under addition is a cyclic group.

 

This is false.  You might try proving that Q is not cyclic---it would be a good exercise.

 

  1. Not every element of every cyclic group generates the group.

 

Right.

 

  1. There is at least one abelian group of every finite order > 0  (there are 2 abelian groups: one with one element and another with 2 elements).

 

I'm not sure what your parenthetical remark means.

 

  1. Every group of order  4  is cyclic.

 

Not true.  The Klein 4-group V is not cyclic.  (p. 59)

 

  1. All generators of Z20 are not prime numbers (9 is a generator and it is not prime).

 

Right.

 

  1. If G and G’ are groups then G G’ is a  group.

 

Subtle point: G and G' might be disjoint.  You can fix this by writing "If G and G' are subgroups of a group H, then G intersect G' is a group."

 

  1. If H and K are subgroups of a group G, then H K is a group.

 

Right.

 

  1. Every cyclic group of order >2 has at least 2 distinct generators.

 

Right.  You might try proving this statement---it would be a good exercise.

 

 

Section 8 - #35.

 

  1. Every permutation is a 1-1 function.

 

Right.

 

  1. Every function is a permutation if it is 1-1 and onto (it cannot be just 1-1).

 

Right.

 

  1. Every function on a finite set onto itself must  be 1-1.

 

False.  Consider a constant function.

 

  1. Every group G may not be isomorphic to a subgroup of SG.

 

Change "may not" to "is." This is Cayley's Theorem.

 

  1. Every subgroup of an abelian group is abelian.

 

Right.

 

  1. Every element of a group may not generate a cyclic subgroup of the group.

 

A better way to phrase this is ". . . it is not necessarily true that every element is a generator." 

 

  1. The symmetric group of S10 has 10! elements.

 

Right.

 

  1. The symmetric group S3 is cyclic.

 

False.  (It's not abelian.)

 

  1. Sn is cyclic for any n.

 

False.  See Section 8 Ex. #46 and Thm. 6.1.

 

  1. Every group is isomorphic to some group of permutations.

 

Right.