Abstract Algebra Group Theory Basics Cheat Sheet

Group theory studies sets with one operation that follows a few strict rules. This cheat sheet helps students recognize when a structure is a group and use the most important results efficiently. It is useful for organizing examples such as integers under addition, nonzero real numbers under multiplication, and symmetry operations.

These ideas build the foundation for higher algebra, number theory, and modern mathematics.

The core definition requires closure, associativity, an identity element, and inverses. Important tools include subgroup tests, cyclic group notation, element order, cosets, and Lagrange’s theorem. Many problems come down to checking whether an operation is well defined and whether the group axioms hold.

Once a group is known, formulas such as $|G| = |H|[G:H]$ give powerful shortcuts.

Key Facts

A group $(G, *)$ is a set $G$ with an operation $*$ that satisfies closure, associativity, identity, and inverses.
Closure means that for all $a,b \in G$ , the result $a*b \in G$ .
Associativity means that for all $a,b,c \in G$ , $(a*b)*c = a*(b*c)$ .
An identity element $e \in G$ satisfies $e*a = a*e = a$ for every $a \in G$ .
An inverse of $a \in G$ is an element $a^{-1} \in G$ such that $a*a^{-1} = a^{-1}*a = e$ .
A group is abelian if $a*b = b*a$ for all $a,b \in G$ .
The order of an element $a$ is the smallest positive integer $n$ such that $a^n = e$ , if such an $n$ exists.
Lagrange’s theorem says that if $H \leq G$ and $G$ is finite, then $|H|$ divides $|G|$ and $|G| = |H|[G:H]$ .

Vocabulary

Group: A group is a set with one binary operation that satisfies closure, associativity, an identity element, and inverses.
Abelian Group: An abelian group is a group whose operation is commutative, so $a*b = b*a$ for all elements.
Subgroup: A subgroup is a subset $H$ of a group $G$ that is itself a group under the same operation, written $H \leq G$ .
Cyclic Group: A cyclic group is a group generated by one element $g$ , meaning every element can be written as $g^n$ for some integer $n$ .
Coset: A left coset of $H$ in $G$ has the form $aH = \{ah : h \in H\}$ for some $a \in G$ .
Order: The order of a finite group is its number of elements, and the order of an element $a$ is the least positive $n$ with $a^n = e$ .

Common Mistakes to Avoid

Forgetting to check closure is wrong because an operation can look valid but produce results outside the set, so it cannot form a group.
Assuming commutativity is automatic is wrong because groups only require associativity, and many important groups have $a*b \neq b*a$ .
Using the wrong identity element is wrong because the identity must satisfy both $e*a = a$ and $a*e = a$ for every element in the set.
Confusing element order with group order is wrong because $|G|$ counts all elements in the group, while $|a|$ counts powers of one element until $e$ appears.
Applying Lagrange’s theorem backward is wrong because $d \mid |G|$ does not guarantee that a subgroup of order $d$ exists.

Practice Questions

1 Determine whether $(\mathbb{Z}, +)$ is a group, and identify its identity element and the inverse of $7$ .
2 In the group $\mathbb{Z}_{12}$ under addition modulo $12$ , find the order of the element $8$ .
3 If $G$ is a finite group with $|G| = 30$ and $H \leq G$ with $|H| = 5$ , find the index $[G:H]$ .
4 Explain why the set of nonzero integers $\mathbb{Z} \setminus \{0\}$ under multiplication is not a group, even though multiplication is associative.

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Abstract Algebra Group Theory Basics Cheat Sheet

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