Normal Subgroups of Prime Power Index
Normal subgroups of prime power index are kernels of surjective maps to p-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem.
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:
- Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian group, and is the largest elementary abelian p-group onto which G surjects.
- Ap(G) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index normal subgroup that contains the derived group ): G/Ap(G) is the largest abelian p-group (not necessarily elementary) onto which G surjects.
- Op(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index normal subgroup): G/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects. Op(G) is also known as the p-residual subgroup.
As these are weaker conditions on the groups K, one obtains the containments
These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.
Read more about this topic: Index Of A Subgroup
Famous quotes containing the words normal, prime, power and/or index:
“When a man says that he is Jesus or Napoleon, or that the Martians are after him, or claims something else that seems outrageous to common sense, he is labeled psychotic and locked up in a madhouse. Freedom of speech is only for normal people.”
—Thomas Szasz (b. 1920)
“Vanessa wanted to be a ballerina. Dad had such hopes for her.... Corin was the academically brilliant one, and a fencer of Olympic standard. Everything was expected of them, and they fulfilled all expectations. But I was the one of whom nothing was expected. I remember a game the three of us played. Vanessa was the President of the United States, Corin was the British Prime Minister—and I was the royal dog.”
—Lynn Redgrave (b. 1943)
“Whatever woman may cast her lot with mine, should any ever do so, it is my intention to do all in my power to make her happy and contented; and there is nothing I can imagine, that would make me more unhappy than to fail in the effort.”
—Abraham Lincoln (1809–1865)
“Exile as a mode of genius no longer exists; in place of Joyce we have the fragments of work appearing in Index on Censorship.”
—Nadine Gordimer (b. 1923)