Direct product is cancellative for finite groups

From Groupprops
Jump to: navigation, search

Statement

Suppose are finite groups, such that:

where denotes the external direct product. (Note that the isomorphism need not

Then, we have:

.

Related facts

Related facts for other algebraic structures

The statement is true for finite algebras in any variety of algebras:

Stronger facts for groups

Facts for other kinds of products

Other related facts

Facts used

  1. Homomorphism set to direct product is Cartesian product of homomorphism sets: If are groups, then there is a natural bijection:
    • .
    • The bijection is defined as: .
  2. Homomorphism set is disjoint union of injective homomorphism sets: For groups and , let denotes the set of homomorphisms from to , and denote the set of injective homomorphisms from to . Then we have:

.

Proof

Given: Finite groups such that .

To prove: .

Proof: Let be an arbitrary finite group.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 as an equality of finite numbers. Fact (1) are finite [SHOW MORE]
2 as an equality of finite numbers. Fact (1) are finite [SHOW MORE]
3 as an equality of finite numbers. The number of homomorphisms to a group depends only on its isomorphism type.
4 as an equality of finite numbers. Steps (1),(2),(3) [SHOW MORE]
5 For any finite group , the number of injective homomorphisms from to equals the number of injective homomorphisms from to . We show this by induction on the order of . In other words, Fact (2) Step (4) [SHOW MORE]
6 is isomorphic to a subgroup of and is isomorphic to a subgroup of are finite Step (5) [SHOW MORE]
7 is isomorphic to are finite Step (6) [SHOW MORE]