Definitions
A set together with a binary operation that has three properties:
1. Associativity
2. Identity
3. Inverses
What is a group?
A subset H of a group G that is itself a group under the operation of G.
What is a subgroup?
Every group is isomorphic to a group of permutations.
This group consists of all the permutations of a set with three elements.
What is S3?
I am a group of order 4, but I am not Z4.
What is Z2xZ2?
The subset of elements in G that commute with every element of G.
What is the center of a group?
The number of elements of a group.
What is the order of a group?
For group elements a and b, (ab)-1=b-1a-1
What is the Socks-Shoes Property?
This group consists of the integers modulo 4 under addition, and has exactly four elements.
What is Z4?
I am a group of order 6, but I am not Z6.
What is S3?
The smallest positive integer n such that gn=e.
What is the order of the element g?
A function from A to A that is both one-to-one and onto.
Let G be a group and let H be a nonempty subset of G. If ab is in H whenever a and b are in H, and a-1 is in H whenever a is in H, then H is a subgroup of G.
What is the Two-Step Subgroup Test?
This group consists of the set of all symmetries of a square, including rotations and reflections.
What is the dihedral group D4?
I am a group of order 8, but I am not Z8.
What is D4?
A 2-cycle
What is a transposition?
A permutation that can be expressed as a product of an even number of 2-cycles.
What is an even permutation?
Every integer greater than 1 is a prime or a product of primes. This product is unique, expect for the order in which the factors appear.
What is the Fundamental Theorem of Arithmetic?
This group consists of the integers relatively prime to 10, under multiplication modulo 10. It has four elements and is abelian.
What is U(10)?
I am a group of order 1.
What is the identity?
A function from a group G to a group H that is a one-to-one, onto, and preserves the group operation.
What is an isomorphism?
An isomorphism from a group G onto itself.
What is an automorphism?
Every subgroup of a cyclic group is cyclic. Moreover, if |<a>|=n, then the order of any subgroup of <a> is a divisor of n; and, for each positive divisor k of n, the group <a> has exactly one subgroup of order k- namely <an/k>.
What is the Fundamental Theorem of Cyclic Groups?
This group consists of the set of all 2×2 invertible matrices with real entries, and the operation is matrix multiplication.
What is the general linear group GL(2, R)?
I am a group of order 12, but I am not Z12 or D6.
What is A4?