代數(shù)結(jié)構(gòu)同態(tài)的方法及應(yīng)用(參考版)
2025-06-29 03:30本頁面
【正文】 但是，即使和不同，他們肯定承受彼此非常相似（）。例如，我們有調(diào)查，對所有置換的X =（1,2,3）的群體。 that is, =1. In this case,the first row and first column of the table merely repeat the listing above, and so we usually omit them. Consider two almost trivial examples of groups: let denote the multiplicative group {1，1}, and let denote the parity group() : 1 11 1:even oddodd evenIt is clear that and are distinct groups；it is equally clear that there is no significant difference between them. The notion of isomorphism for malizes this idea。附錄 Y 外文原文 HOMOMORPHISMSAn important problem is determining whether two given groups and are somehow the same. For example, we have investigated, the groups of all permutations of X ={1,2,3}. The groups of all the permutations of Y ={} is a group different from because permutations of {1,2,3} are different than permutations of {}. But even though and are different, they surely bear a strong resemblance to each other (see example ). The notions of homomorphism and isomorphism allow one to pare different groups, as we shall see.Definition . If and are groups (we have displayed the operation in each), then a function f: is a homomorphism if f()=f()f()for all ,. If f is also s bijection, then f is called an isomorphism. Two groups and are called isomorphic, denoted by , if there exist an isomorphism f: between them.For example, let be the group of all real numbers with operation addition, and let be the group of all positive real numbers with operation multiplication. The function f: , defined by f()=, is a bijection [its inverse function is g()=ln]. Moreover, f is an isomorphism, for if , then f(+)===f()f().Therefore, the additive group is isomorphic to the multiplicative group . As a second example ,we claim that the additive group of plex Numbers is isomorphic to the additive group [see example ]. define f: by f: +(,).It is easy to check that f is a bijection。那么f是一個同構(gòu)說，如果我們疊加的乘法表 (取決于,…, )根據(jù)乘法表 (取決于f(),f(),…, f(),然后匹配表: 如果 是輸入在給定的乘法表 , 則 f()f()=f() 是 ，同構(gòu)的群具有相同的乘法表。有一階群的許多乘法表，對每一個!的列表元素。 : 1 11 1:even oddodd even顯然群 和 是兩個不同的群；同樣他們之間不存在顯著差異。 在這個例子中, 第一行和表的第一列只是重復(fù)上面的元素，所以我們通常忽略他們。 例2,我們說所有復(fù)數(shù)對于普通加法作成的加群與加群 同構(gòu)[]. 定義f: f: +(,).這很容易驗證f是雙射； 則說f是一個同態(tài)映射，因為f([+]+[+])=f([+]+[+])=(+,+)=(,)+(,) = f(+)+f(+).定義 . 設(shè),…, 是一組沒有重復(fù)元素的集合構(gòu)成的群。設(shè)f: , 記為 f()=, 是一個雙射 [它的反函數(shù)是g()=ln]。 設(shè) 和 是兩個群, 如果存在一個同構(gòu)映射f：，則稱 和 同構(gòu)，記為。如果f()=f()f()對于任意的,都成立。提供更好的翻譯建感謝您為 Google 翻譯提供翻譯建議。n d224。ntǐ, wǒmen jiāng hu236。t243。 bǐji224。ng g242。ng
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