Question

Give an example of each object described below, or explain why no such object exists:

1. A group with 11 elements that is not cyclic.

2. A nontrivial group homomorphism f : D8 −→ GL2(R).

3. A group and a subgroup that is not normal.

4. A finite integral domain that is not a field.

5. A subgroup of S4 that has six elements.

Answer #1

1.the statement is false.

Because, we know that, every group of prime order is cyclic. Since, 11 is cyclic hence it is always cyclic group. So there exist no group of order 11 that is not cyclic.

3.)yes, such a example exists.

Consider the Symmtric group S3 whose order is 6. It has a subgroup of 2, namely H={e,a}. But it is not normal.

4) No, such a example does not exist.

Because, every finite integral domain is always field.

5).No.such a result is not exist. Because S4 has no subgroup of order 6.

Because converge of Lagranges theorem is not true. This is true only when the group is finite abelian group. But S3 is non - abelian.

Give an example of the described object or explain why such an
example does not exist.
•An orthogonal linear transformation T:
R2→R2.
•An orthogonal linear transformation T:
R3→R3.
•A basis B for R2 and an orthogonal linear
transformation T: R2→R2 such that
[T]B is an orthogonal matrix.
•A basis B for R2 and an orthogonal linear
transformation T: R2→R2 such that
[T]B is NOT an orthogonal matrix.
•A non-orthogonal linear transformation that takes an orthogonal
basis to an orthogonal basis.

For each problem below, either give an example of a function
satisfying the give conditions, or explain why no such function
exists.
(a) An injective function f:{1,2,3,4,5}→{1,2,3,4}
(b) A surjective function f:{1,2,3,4,5}→{1,2,3,4}
(c) A bijection f:N→E, where E is the set of all positive even
integers
(d) A function f:N→E that is surjective but not injective
(e) A function f:N→E that is injective but not surjective

explain why or not. Determine whther the ff statements are true
or not and give an explaination or counter example
1.The vector field F=<3X^2,1> is a gradient field for both
f(x,y)=x^3+y and f(x,y)=y+x^3+100
2.the vector field F=(y,x)/sqrt(x^2+y^2) is constant in
direction and magnitude on the unit circle.
3.the vector field F=(Y,X)/SQRT(X^2+Y^2) IS NEITHER RADICAL
FIELD NOR A ROTATION FIELD.explain
4.If a curve has a parametric description
r(t)=<x(t),y(t),z(t)>, whrer t is the arc length then
magnitude of r'(t)=1.explain
5.the vector field...

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