不可数非对易Specker现象的失效
The failure of the uncountable non-commutative Specker Phenomenon
摘要：
Higman在1952年证明了每个自由群都是非交换细长群，也就是说，如果G是一个自由群，h是可数完全自由积（X_omega Z）到G的同态，则存在omega的一个有限子集F和一个同态h：*_{i in F} Z --> G使得h=h rho_F，其中rho_F是从（X_{i in omega}）Z到 *_{i in F}Z的自然映射。对应于阿贝尔的情况下，这种现象被称为非交换斯派克现象。在本文中，我们表明，Higman的结果失败，如果一个通过从可数到不可数。特别地，我们证明了对于非平凡群G_alpha（alpha在lambda中）和不可数基数lambda，从G_alpha的完全自由积到整数环有2个^{2 ^lambda}同态.
英文摘要：
Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product (X_omega Z) to G, then there exists a finite subset F of omega and a homomorphism h:*_{i in F} Z --> G such that h=h rho_F, where rho_F is the natural map from (X_{i in omega})Z to *_{i in F}Z . Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman's result fails if one passes from countable to uncountable. In particular, we show that for non-trivial groups G_alpha (alpha in lambda) and uncountable cardinal lambda there are 2^{2^lambda} homomorphisms from the complete free product of the G_alpha 's to the ring of integers.
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