Group homomorphism and abelian groups

Reference no: EM13124379

Group Homomorphism and Abelian Groups
Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi).

proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel, but I'm not sure about how to do the reverse (<=) and show that phi is abelian?

Reference no: EM13124379

Questions Cloud

Estate tax purposes as of which date : If an election is available and is made to use alternate valuation for federal estate tax purposes, then if a parcel of real estate owned by the decedent is sold within six months after the decedent's death, the parcel of real estate is valued for..

Illustrate what is the material purchase price variance : Direct material 9 feet at $5.00 per foot. Actual information for July: 2,000 units were produced; 20,000 feet of direct material were purchased at a cost of $92,000 and 19,000 feet of material were used. Illustrate what is the material purchase pr..

Algebra-linear equations and inequalities : The surface area S of a right prism is given by S = 2B + Ph. The length of a rectangle is five times its width. If the area of the rectangle is 500m², find its perimeter.

Group homomorphism and abelian groups : Group Homomorphism and Abelian Groups, Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi).

For profit and government organization : What are the differences in financial reporting in a for-profit and government organization? What are the similarities in financial reporting in a for-profit and government organization?

Which is accurate with respect to the federal estate tax : Under Carl's will, Carl created a testamentary trust to be funded with $700,000 worth of assets. All of the income of the trust is payable to Carl's child, Jane, for her life, and thereafter, the remaining assets of the trust will pass to The Publ..

An academic or professional environment : Why do you think ethics are of concern in an academic or professional environment? How might you determine your personal ethics?

Converting linear equations to slope-intercept form : Write the two equations in slope-intercept form, and then determine how many solutions the system has.

User Account

All Pages