[n] means the topic is found on acetate n.

**Definitions**

Group [2]

Inverse element [2]

Abelian group [20]

nonAbelian group [20]

Examples:

**Z _{n }**under + [25]

Cyclic groups [29 to 33.2]

Groups of small order, like 1, 2, 3, 4, 5 [39, 40, 47, 48, 49]

Subgroup (g) generated by an element g. [29]

Cyclic group [32]

Order of a group [38]

Order of a group element [38]

Function

Onto

One-to-one

Permutation of a set [3]

transposition (2-cycle) [7]

even permutation [8.9]

odd permutation [8.9]

Isomorphism of groups [24]

Subgroup [27.0, 27.1]

Homomorphism of groups [57]

Kernel of a homomorphism [61.3]

Quotient group G/H of a group G over a normal subgroup H. [67 to 70]

Symmetry of a geometric figure [75]

Every permutation is a product of transpositions [7 to 8.0]

The identity element in a group is unique [20]

The inverse of a given group element is unique [21]

(a')' = a [21]

(ab)' = b'a' [22]

If ab = ac, then b = c [22]

If ba = ca, then b = c [22]

If a is a fixed element of a group G, then the function F_{ }which
maps g in G to ga, is a permutation of G. [ 23]

If H is a subgroup of G, then the identity element of g is in H. [27.1]

If H is a nonempty subset of a group G with the property that for every h, k in H, hk' is in H, then H is a subgroup of G. [28]

The set of all powers of a fixed element g is a subgroup. [29]

A cyclic group is Abelian [30]

If g^{k} = g^{m }for different exponents k and m, then
(g) is a finite cyclic group and the number of elements in (g) is equal
to the least positive integer n for which g^{n} = e. [31]

If H is a subgroup of G, then

If a and b are in G, then aH and bH are either disjoint or equal (as subsets of G) (Same for Ha and Hb.)

All cosets, left and right have the same number of elements. [44 to
46.0]

If F is a homomorphism from G into C, then

F(a') = F(a)' for every a in G [61.0]

A homomorphism is one-to-one if and only if its kernel is {e}. [65]

If H is a normal subgroup of a group G, the mapping F from G onto G/H
given by F(a) = aH is a homomorphism with kernel H. [71.0, 71.1]

Every permutation is either even or odd, but not both [8.9 to 14]

Every cyclic group is either infinite and isomorphic to **Z**, or
is finite and, if it has n elements, is isomorphic to **Z _{n}**.
[30 to 33.0]

Every subgroup of a cyclic group is cyclic [34 to 35]

Know what are all the subgroups of a cyclic group [36 to 37.1]

Every group of prime order p is isomorphic to **Z _{p}**,
+ [49]

Every group of order 4 isomorphic to either **Z _{4}** or

Every group of order 6 isomorphic to either **Z _{6}** or

If H is a subgroup of G, then aH = bH if and only if a'b is in H. [56.4]

If F is a homomorphism from G into C, then

if H is a subgroup of G, then F(H) is a subgroup of C. [61.1]

A normal subgroup is the kernel of some homomorphism. [66]

A symmetry of a polygon or polyhedron is completely determined by how it permutes the vertices. [77]

The set of symmetries of a geometric figure is a group with multiplication
equal to composition of functions. [79, 80]

Calculate products using cycle notation [6]

Write any permutation as a product of transpositions [7 to 8.0]

Determine whether a permutation is even or odd [8.9]

Determine whether or not a given configuration of the fifteen puzzle
can be reached from the initial configuration. [16 to 19]

a' or a^{-1} for the inverse of a [2]

composition of functions, read left to right [3]

aF for F(a) [4.1, 4.2]

Two-row notation for permutations [5]

Cycle notation for permutations [5.0]

Notation for isomorphism [25]

(g) for the cyclic subgroup generated by g. [29]