### Groups : Isomorphism and Homomorphism

Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

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Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

Note: S4 means symmetric group of degree 4 A4 means alternating group of degree 4 e is the identity Is there a group homomorphism $:S4 -> A4, with kernel $ = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}?

Show that innG is the normal subgroup of autG for any group G Note: innG = inner automorphism group of G aut G = automorphism group of G

If G = <X> and $:G->G1 is an onto homomorphism, show that G1 = <$(X)>, where $(X) = the set of $(x) given that x belongs to X.

Note: ~~ means an isomorphism exists. Moreover,if an isomorphism existed from G to G1 I would say G ~~ G1 Questions: If G is an infinite cyclic group, show that G ~~ Z (Z is the set of integers)

The attached file has some slides provided by my professor on the univariate method and powell's method. I am having trouble understanding, so I tried to work an example, but I am not getting very far. As you work the example, could you explain each step as you go. My professor tried but he and I both ended up frustrated, a

** Please see the attached file for full problem description ** Let T be a linear operator on P_3 defined as follows: T(ax^3 + bx^2 + cx + d) = (a - b)x^2 + (c - d)x + (a + b - c). The matrix [T]_G which represents T with respect to the basis G which = {1 + x, 1 - x, 1 - x^2, 1 - x^3}. Show that

Please see the attached file for the fully formatted problems. Suppose  is an onto homomorphism from ℤ16 to a group G of order 4. Find ker(). Explain your answer.

Extension of A_5

Let a be a fixed vector in R2. A mapping of the form L(x) = x+a is called a translation. Show that if a does not equal 0, then L is not a linear transformation. Describe or illustrate geometrically the effect of the translation. Thanks for your help!

See attached file for full problem description with symbols and equations. --- Definition 11.1 An orthogonal projection operator is a linear transformation such that and . Question: If W is a subspace of V, prove that P_w is an orthogonal projection. (P_w is P sub w)

Please see attached file.

Please see attached file

Is the following transformation linear? T: R^3 -> R^2 defined by T(x,y,z)=(x,y)