Let be a group, a subgroup, and a fixed element. Prove that if for some element , then and .
Suppose for some . In particular, ; then for some , so that . Left multiplying by we have , but since , , and we have . Moreover, so that .
Let be a group, a subgroup, and a fixed element. Prove that if for some element , then and .
Suppose for some . In particular, ; then for some , so that . Left multiplying by we have , but since , , and we have . Moreover, so that .
On these pages you will find a slowly growing (and poorly organized) list of proofs and examples in abstract algebra.
No doubt these pages are riddled with typos and errors in logic, and in many cases alternate strategies abound. When you find an error, or if anything is unclear, let me know and I will fix it.
Comments
Why is k an element of Hg? I agree that if kH=gH, then k is an element of gH (or for right cosets), but why is this true for “mixing” left and right cosets?
What is a coset?
is the set of all products of the form where . Now let .