If G and H are abelian prove that G times H is abelian Suppo

If G and H are abelian, prove that G times H is abelian Suppose the groups G and H both have the following property: Every element of the group is its own inverse. Prove that G times H also has this property. Powers and Roots of Group Elements Let G be a group, and a, b G. For any positive integer n we define a\" by If there is an element x G such that a = x^2 we say that a has a square root in G Similarly, If a = y^3 for some y G, we say a has a cube root in G. In general a has an n the root in G if a = z^n for some z G. Prove the following: (bab^-1)^n = ba^n b^-1 for every positive integer n. Prove by induction. (Remember that to prove a formula such as this one by induction, you first prove it for n = 1; next you prove that if it is true for n = k then it must be true for n = k +1. You may conclude that it is true for every positive integer n Induction is explained more fully in Appendix C.) If ab = ba then (ab)^n = a^n b^n for every positive integer n Prove by induction. If xax = e, then (xa)^2n =a^n. If a^3 = e, then a has a square root. If a^2 = e, then a has a cube root. If a^-1 has a cube root so does a. If x^2 ax = a^-1, then a has a cube root If xax = b, then ab has a square root.

Solution

Ans-

o perform an elementary column operation on A, an r x c matrix, take the following steps.

Let\'s work through an elementary column operation to illustrate the process. For example, suppose we want to interchange the first and second columns of A, a 3 x 2 matrix. To create the elementary column operator E, we interchange the first and second columns of the identity matrix I₂.

Then, to interchange the first and second columns of A, we postmultiply A by E, as shown below.

Note that the process for performing an elementary column operation on an r x c matrix is very similar to the process for performing an elementary row operation. The main differences are:

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1 0 0 1		0 1 1 0
I2		E