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Set equipped with a single operation that combines any two elements to form a third element, satisfying four conditions: closure, associativity, identity, and inversibility.

Set with two binary operations (addition and multiplication) satisfying certain conditions, including: a group under addition, distributive property of multiplication over addition, and multiplication is associative.

Set on which two operations, addition and multiplication, are defined and behave as they do with rational and real numbers, including the existence of additive and multiplicative inverses (except for the additive identity).

A function between two groups that preserves the group operation: $f(a * b) = f(a) * f(b)$ for all $a, b$ in the group.

A subgroup $N$ of a group $G$ such that for all elements $n$ in $N$ and $g$ in $G$, the element $gng^{-1}$ is also in $N$.

A subset of a ring that is itself a subring and is closed under multiplication by elements from the ring.

The group consisting of the cosets of a normal subgroup, with the operation defined by the product of cosets.

A collection of vectors that can be scaled and added together to produce another vector in the same space, satisfying axioms related to vector addition and scalar multiplication.

A generalization of a vector space, except the scalars come from a ring instead of a field.

A bijective group homomorphism; a structure-preserving one-to-one correspondence between two groups.

A function between two rings that respects both the addition and multiplication operations of the rings.

A larger field containing a smaller field as a subfield; algebraic structures important in the study of field theory and Galois theory.

The set of all products of a particular element from the group with all of the elements of a particular subgroup.

The group constructed from two groups, whose elements are ordered pairs from the two groups, with the group operation defined componentwise.

A nontrivial group whose only normal subgroups are the trivial group and the group itself.

The set of elements in the domain of the homomorphism that are mapped to the identity element of the codomain.

A group that can be generated by a single element, meaning that every element in the group can be written as some power of this generator.

A group where the group operation is commutative, meaning that for any two elements $a$ and $b$ in the group, $ab = ba$.

A subset of a ring that is itself a ring with the same addition and multiplication as the larger ring.

The smallest positive number of times the identity under addition must be added to itself to get zero; if there is no such number, the ring has characteristic zero.

A ring that also has the structure of a vector space over a field, together with a bilinear product.

A group formed by the set of all possible words that can be built from members of a given set, called generators, subject to the only relations being that a generator composed with its inverse gives the identity element.

The group of field automorphisms (bijection from a field to itself that preserves the algebraic operations) that leave a certain smaller field fixed, important in understanding the solvability of polynomial equations.

Given two elements $a$ and $b$ of a group, the commutator is the element $[a,b] = aba^{-1}b^{-1}$.