Abstract Algebra: Identity Element of the Group
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1. Prove that is a is a number in G, a group, and ab = b for some b of G, then a = e, the identity element of the group.
2. Consider the set of polynomials with real coefficients. Define two elements of this set to be related if their derivatives are equal. Prove that this defines an equivalence relation.
3. Let H be a subgroup of the group G. Prove that every right coset of H is a left coset of some subgroup of G.
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Solution Summary
This solution helps answer various abstract algebra questions. It helps provides proof for a number, identifies elements of a group, defines elements of a set, and proves cosets of a subgroup.
Solution Preview
1. The element b has a unique inverse, b, such that bb=bb=e . Hence
ab=b
(ab)b^?1=bb^?1
a(bb^?1)=bb^?1 (associativity of group multiplication)
ae=e (definition of inverse)
a=e (identity property of e )
2. Consider the set of polynomials with real coefficients. Define two elements of this set to be related if their derivatives are equal. Prove that this defines an equivalence ...
Purchase this Solution
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