Exploring Group Homomorphisms from the Baer-Specker Group to the Integers

The study of group homomorphisms from the Baer-Specker group to the integers, denoted as ( phi: G/F to mathbb{Z} ), provides deep insights into the structure of the Baer-Specker group and its relationship with free abelian groups. This article delves into the unique properties and implications of such homomorphisms, culminating in a significant result that highlights the group's behavior.

The Baer-Specker Group and its Subgroup

The Baer-Specker group, denoted as ( G prod_{n in mathbb{N}} mathbb{Z} ), is the unrestricted product of countably many copies of ( mathbb{Z} ). An element of this group can be represented as an infinite sequence of integers: ( a_0a_1a_2ldots ). A subgroup ( F oplus_{n in mathbb{N}} mathbb{Z} ) of ( G ) consists of elements with only finitely many non-zero entries. These groups provide a rich field for studying group homomorphisms.

Understanding Homomorphisms from ( G/F ) to ( mathbb{Z} )

A homomorphism ( phi: G/F to mathbb{Z} ) is the same as a homomorphism ( phi: G to mathbb{Z} ) that vanishes on ( F ). This means that ( phi(g) phi(g) ) whenever ( g ) and ( g' ) differ only in finitely many places. Essentially, elements that are equivalent under the equivalence relation defined by differences in finitely many places are mapped to the same integer by the homomorphism.

A Fundamental Observation

A key observation is that if ( g a_0a_1a_2ldots ) is an element of ( G ) and there exists a natural number ( d ) such that ( a_k ) is divisible by ( d ) for all ( k ), then ( phi(g) ) must be a multiple of ( d ). This is because ( g ) can be expressed as the sum of ( d ) identical elements, each of which is ( a_0/d, a_1/d, a_2/d, ldots ).

Divisibility by Unbounded Powers

Consider an element ( g a_0a_1a_2ldots ) in ( G ) where ( a_k ) is divisible by ( 2^k ) for all ( k in mathbb{N} ). For instance, ( g ) might be ( 761288096ldots ). The entries are divisible by higher and higher powers of 2. The argument proceeds as follows:

For any natural number ( n ), modify ( g ) to ( g 00ldots0a_na_{n1}ldots ) by zeroing out the first ( n ) entries, leaving ( a_n ) as the first potentially non-zero entry.

In this new element ( g ), every entry is divisible by ( 2^n ). Since ( g ) and ( g ) are equivalent, ( phi(g) phi(g) ).

Since ( phi(g) ) is divisible by ( 2^n ) for every ( n ), it follows that ( phi(g) 0 ). The same argument applies to elements whose entries are divisible by unbounded powers of 3.

Finally, for an arbitrary element ( x x_1x_2ldots ) in ( G ), any ( x_n ) can be expressed as ( x_n u_n 2^n v_n 3^n ), where ( u_n ) and ( v_n ) are integers. Thus, ( x ) can be written as the sum ( x g h ) of two elements where ( phi(g) 0 ) and ( phi(h) 0 ). Therefore, ( phi(x) phi(g h) 0 ).

This demonstrates that under the given conditions, ( phi ) maps any element of ( G ) to 0, making ( phi equiv 0 ).

Conclusion

The results presented above highlight the unique behavior of the Baer-Specker group and its subgroups. This study not only enriches our understanding of group theory but also provides a foundation for further exploration into the structure of infinite groups and their homomorphisms.