Powering-injectivity is central extension-closed
This article gives the statement, and possibly proof, of a group property (i.e., powering-injective group for a set of primes) satisfying a group metaproperty (i.e., central extension-closed group property)
View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties
Get more facts about powering-injective group for a set of primes |Get facts that use property satisfaction of powering-injective group for a set of primes | Get facts that use property satisfaction of powering-injective group for a set of primes|Get more facts about central extension-closed group property
- is -powering-injective, i.e., is injective from to itself.
- The quotient group is -powering-injective, i.e., is injective from to itself.
Then, the whole group is -powering-injective.
Given: A group , a prime number . A central subgroup of such that in both and viewed separately, the map is injective. Two elements with .
To prove: .
Proof: Let be the quotient map.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||.||, and the map is injective in .||We have , hence . Thus, , so are elements of with the same power. Thus, they must be equal.|
|2||The element is an element of .||Step (1)||is the identity element of , so .|
|3||is the identity element of .||, is central in .||Step (2)||We have . Thus, . By the centrality of , we get math>a^p = u^pb^p</math>. We also have . This gives , so is the identity element of , and hence also of the subgroup .|
|4||is the identity element of .||is injective in .||Step (3)||Step-given direct.|
|5||.||Steps (2), (4)||Step-combination direct.|