Cardinality, Products, Subgroups and Homomorphisms — Studies in Algebra and Group Theory

N4 Cardinality, Products, Subgroups and Homomorphisms

5 September 2025

Set theory review

Hook right arrow (injective): we may write .
Surjective: we may write .
Bijective: .

Definition N4.1 (Cardinality). We say and have the same cardinality if there exists a bijection , and write . If there exists an injection , .

Theorem N4.2 (Schröder–Bernstein). If there exist injective and , then there exists a bijection , so . (Full proof: Halmos p. 88.)

For , notice: send , where , if , and , where , if .

: countably infinite.

Theorem N4.3 ( is uncountable). No map can be surjective.

Proof. Cantor’s diagonal argument: use the binary or decimal expansion. Write Let where for all (along the diagonal). Then for all . ∎

“Real mathematicians don’t care about the axiom of choice, get over it!” (Prof. Auroux, 2025)

Using the same diagonal argument we know there are arbitrarily large cardinals. Easy algorithm for finding them: take any set and find ; then , where (margin: confused; wtf Cantor). If is finite, . If is infinite, (proof by Cantor):

Proof. Given , we show it cannot be surjective. Let Then for any , , so . ∎

Back to group theory

(Products of countably infinite sets, say , have the same cardinality as : .)

Definition N4.4 (Product of groups). Given groups and , with , where is the operation on and the two are the operations of and of . If , , then .

Products of more than 2 groups: .

Given infinitely many groups (margin: understand more):

Definition N4.5 (Direct product, direct sum). (Margin: this is why we don’t use .)

Example N4.6. Consider and denote by : finitely many terms give polynomials, infinitely many give power series. So

Subgroups

Definition N4.7 (Subgroup). A subgroup of a group is a non-empty subset of the set which is closed under the operation (law of composition), , and closed under inversion, . These two properties imply : notice and . with the restricted operation of is a group. Say is a proper subgroup if . Say is the trivial subgroup if .

Proposition N4.8. If are subgroups, then is also a subgroup of .

(Margin: if you take the intersection of all subgroups, does it result in the trivial subgroup?)

Homomorphisms

Definition N4.9 (Homomorphism). Given groups and , a homomorphism is a map which respects the operation (law of composition): for , This implies and .

Remark. A pedantic way to state is by using a commutative diagram: Going around either way gives the same element: the two results are equal.

Definition N4.10 (Isomorphism). An isomorphism is a bijective homomorphism.

Example N4.11.

All groups of order 2 are isomorphic to each other: .
.
(the symmetries of a triangle with vertices labelled ).
.
, (homomorphism but not isomorphism).
If , ; think (homomorphism, not isomorphism).

Definition N4.12 (Kernel). The kernel of a group homomorphism is the set of all elements of which go to the identity element of : It is a subgroup of .

is injective : Therefore the kernel also shows the “difference” of two elements which map to the same element (identity).