[Abstract Algebra] Chapter 17 Flashcards
What is a Sylow p-subgroup?
A subgroup of order p^n in a group G where v_p(|G|)=n and |G|=m(p^n)
What does the variable n_p represent in the Sylow Theorem?
The number of Sylow p-subgroups of G
What does the Sylow Theorem state in relation to divisibility?
m|n_p
What does the Sylow Theorem state in relation to modular arithmetic?
n_p is congruent to 1 mod p, implying that n_p is nonzero
What does the Sylow Theorem state in relation to conjugate subgroups?
Any two Sylow p-subgroups are conjugate subgroups, therefore isomorphic
What is a conjugate subgroup?
Subgroups H and K of G such that there exists an element g in G satisfying G = gKg^(-1)
What important corollary of the Sylow Theorem is related to normal subgroups?
A Sylow p-subgroup if normal iff n_p = 1
What important corollary of the Sylow Theorem is related to Abelian groups?
Any Abelian group has exactly one Sylow p-subgroup for every p dividing its order
What important corollary of the Sylow Theorem is related to semiprime numbers?
For a group with order pq, n_q = 1
What important corollary of the Sylow Theorem is related to the intersection of groups?
The intersection of a Sylow p- and q- subgroup is the trivial group, when p != q
What is a simple group?
A group with no normal subgroups other than itself and the trivial group
What is the analogous concept to the simple group in elementary number theory?
Prime numbers
What is a composition series?
A sequence of subgroups, {1} = H0 a subgroup of H1, a subgroup of H2, …, a subgroup of Hn = G
What are composition factors?
The groups H1/H0, H2/H1, … in a composition series
What does the Jordan-Holder theorem state?
Every finite group admits a unique composition series up to equivalence
