Abstract Algebra 1

Homomorphisms and Automorphisms

100

First isomorphism theorem says G/Ker(Φ) is isomorphic to this group

What is Φ(G)?

200

S₃ has this many automorphisms

What is 6?

300

The Inner Automorphism Group of Z_n

What is U_n?

400

If G is finite, the inverse map (f(x)=x^-1) is an automorphism if and only if this property is true

What is abelian?

500

If G is finite with a subgroup H of index 2, every automorphism must do this to H.

What is map H to itself?

Cayley's Theorem and ISH

100

The representation of an element g in G as a permutation following Cayley’s Theorem

What is left multiplication permutation

200

Cayley's Theorem says that any group of order n can be embedded in this symmetric group

What is S_n

300

In Cayley's Theorem, if G has order n, then this is the maximum possible size of the image of G within the embedding into the symmetric group

What is n! ?

400

Following ISH, if we have a homomorphism from G to a the symmetric group of the set of right cosets of a subgroup H, this is the kernel of the homomorphism.

What is the largest normal subgroup of G contained in H

500

ISH lets us cut down the size of S for our homomorphism from G to A(s) when this is true

G has no non-trivial normal subgroups

Permutation Groups

100

In the symmetric group S_n, permutations are classified as either even or odd based on this property

What is parity?

200

Every permutation is a product of this

2-cycles

300

The order of an n-cycle

What is n?

400

The index of A_n in S_n

What is 2?

500

Two cycles commute if they have this property

What is disjoint?

Subgroups and Quotient Groups

100

The equivalence relation that forms cosets of a subgroup H

What is ab^-1 in H

200

This is equivalent to left and right cosets being equal

What is normality?

300

This needs to be true for HK to be a subgroup

What is HK=KH?

400

The size of HK

o(H)o(K)/o(H∩K)

500

This subgroup forms the largest abelian quotient group

What is the commutator group?