Definitions
The number of distinct left cosets of H in G.
What is the index?
A map from a group G to a group H that preserves the group operation.
What is a homomorphism?
If G is a finite group and H is a subgroup of G, then |H| divides G. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|.
What is Lagrange's Theorem?
For group elements a and b, (ab)^-1=b^-1a^-1
What is the Socks-Shoes property?
A commutative ring with characteristic 6.
What is Z6?
A subgroup H of a group G with the property that aH=Ha for all a in G.
What is a normal subgroup?
Let phi be a map from a group G to a group H with identity e. This is the set of elements x in G that map to the identity in H.
What is the kernel?
Let G be a finite group of permutations of a set S. Then, for any i from S, |G|=|orb_G(i)|*|stab_G(i)|.
What is the Orbit-Stabilizer Theorem?
A nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication.
What is the Subring Test?
A field with characteristic 11.
What is Z11?
The least positive integer n such that nx=0 for all x in R.
What is the characteristic?
A nonzero element in a ring that is an identity under multiplication.
What is a unity?
A subgroup H of G is normal in G if and only if xHx^-1 is a subset of H for all x in G.
What is the normal subgroup test?
Every subgroup of a cyclic group is cyclic. Moreover, if |<a>|=n, then the order of <a> is a divisor of n; and, for each positive divisor of k of n, the group <a> has exactly one subgroup of order k- namely <a^n/k>.
What is the Fundamental Theorem of Cyclic Groups?
A ring without unity.
What is 2Z?
A nonzero element of a commutative ring with a multiplicative inverse.
What is a unit?
A commutative ring with unity in which every nonzero element is a unit.
What is a field?
Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an element of order p.
What is Cauchy's Theorem for Abelian Groups?
If M and N are normal subgroups of G and N<M, (G/N)/(M/N) is isomorphic to G/M.
What is the Third Isomorphism Theorem?
An integral domain with characteristic zero.
What is Z?
A commutative ring with unity and no zero divisors.
What is an integral domain?
A nonzero element a of a commutative ring R such that there is a nonzero element b in R with ab=0.
What is a zero divisor?
For every integer a and every prime p, a^p mod p=a mod p.
What is Fermat's Little Theorem?
Let phi be a group homomorphism from G to H. Then the mapping G/ker(phi) to phi(G) is an isomorphism.
What is the First Isomorphism Theorem?
A commutative ring with zero divisors.