Hadamard群的研究源于研究Hadamard矩阵，即满足所有不同行的内积为0的{1,-1}-矩阵。1893年Hadamard证明了这种矩阵的阶必然为1,2或者是4的倍数, 后称这类矩阵为Hadamard矩阵。反之， Hadamard曾猜想：如果n为4的倍数，则一定存在n阶 Hadamard矩阵。此猜想至今未解决。1994年Ito给出Hadamard群的定义(见正文)，并证明了如果存在8n阶Hadamard群，则存在4n阶的Hadamard矩阵。之后, Ito猜想群G(n)=〈a,b|a^{4n}=1, a^{2n}=b^2, a^b=a^{-1}〉为Hadamard群。本项目研讨Hadamard 群与扩张函数之间的关系，得到了一个新等价条件，将寻找Hadamard 子集问题转化为整数上的分配格的问题，由此研究Ito 猜想及其Hadamard 群的结构分类，这是其他学者没有尝试过的方法。

The Hadamard group is original from research on Hadamard matrix. In 1867, Sylvester studied the matrices with entries in {−1, 1} such that any two distinct rows have inner product 0. In 1893, Hadamard proved that the order of such matrix was 1,2 or a multiple of 4, which was called Hadamard matrix. Conversely, Hadamard conjectured that if n is a multiple of 4, then there exists an Hadamard matrix of order n. By far it is still open. In 1994, Ito gave a definition of Hadamard groups: Let G be a finite group with the central element -1 of order 2. We call G an Hadamard group if there exists a transversal D of the subgroup〈-1〉in G such that the equlity |D∩Dx|=|G|/4 holds for all elements x outside {1,-1}. He also proved that the Hadamard matrix of order 4n existed if there was an Hadamard group of order 8n. After that Ito conjectured the groups G(n)=〈a,b|a^{4n}=1,a^{2n}=b^2, a^b=a^{-1}〉were Hadamard for all n. The project will fist focus on the relation between extension functions and Hadamard groups. We give an equivlent condition to find out the Hadamard subset by using of a distribute lattice of the integer ring. Then we will use this equivlent condition to research Ito's conjecture and classify the structure of Hadamard groups, which is the method that other researchers don't use before.