Endomorphism kernel implies quotient-powering-invariant
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., endomorphism kernel) must also satisfy the second subgroup property (i.e., quotient-powering-invariant subgroup)
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Suppose is a group and is an endomorphism kernel in , i.e., is a normal subgroup of and there is a subgroup of such that . Then, is a quotient-powering-invariant subgroup of , i.e., for any prime number such that is -powered, so is the quotient group .
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Given: A group , a normal subgroup of such that is isomorphic to a subgroup of . is powered over a prime number .
To prove: is powered over .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||Let be the composite of the quotient map and the isomorphism from to .||is normal in and .|
|2||For , there exists a unique such that .||is powered over .|
|3||The obtained in Step (2) is actually an element of . In other words, for , there exists a unique such that . So, is powered over .||Steps (1), (2)||Suppose (note that such a exists because the image of is ). Then, since , there exists such that . Thus, , with . By the uniqueness of from Step (2), . Thus, .|
|4||is powered over .||.||Steps (1), (3)||Step (1) says that . Step (3) says that is powered over . Thus, is powered over .|