单位元群

Let G be a finite group, and S a subset of G not containing theidentity element 1. The Cayley digraph Cay on G with respect to S is definedas a directed graph with vertex set G and edge set {|g∈G, s∈S}. IfS~(-1)=S, then Cay is undirected and in this case, we view two directededges and as an undirected edge {u, v}. Obviously, the right regularrepresentation R of G, the acting group of G by right multiplication, is asubgroup of the full automorphism group AutCay(G, S of Cay.

给定有限群G，设S是G的不含单位元1的子集，群G关于子集S的Cayley有向图Cay是一个以G顶点集合，而以｛|g∈G，s∈S｝为边集合的有向图，特别地，若S~(-1)=S，则X=Cay是无向的，此时我们把一条无向边｛u，v｝等同于两条有向边和，易见，群G的右正则表示R，即G在G上的右乘作用，为图Cay的全自同构群AutCay

Conversely, if ''S'' is a subgroup of ''G'', then it obeys the axioms of a group.

因此，满足了闭包、单位元和逆元公理，而结合律是继承来的，所以''S''是子群。

A language is called an identity biordered language over A,shortened as IBL,if it is recognized by the identity element of a monoid.

语言称为A上的单位语言，若它被某么半群的单位元所识别。

Let S_n = Sym be the symmetric group on the set {1, 2,…, n}, T_n is a set of identity element and all transpositions of Sym.

设S_n=Sym是集合{1，2，…，n上的对称群，T_n是由对称群的单位元和所有对换构成。

Simons [30] proved the non-existence theorem for stable integral current in acompact Riemannian submanifold isometrically immersed into a unit sphere andvanishing theorem for homology groups. In 1984, Y. L. Xin [47] generalized theLawson-Simon\'s nonexistence theorem for stable integral current and vanishingtheorem for homology groups to the case of compact submanifolds in Euclideanspace, and gave several important applications.

Simons运用Federer-Fleming存在性定理[19]和几何测度论中变分技巧证明了单位球面中紧致黎曼子流形上稳定积分流的不存在性定理和同调群消没定理[30]。1984年，忻元龙将Lawson-Simons稳定积分流的不存在性定理和同调群消没定理拓广到了欧氏空间中紧致子流形的情形，并给出了若干重要的应用[47]。1997年，K。

Furthermore, when the normal subgroup is the idendity group, the basis of the first space is just as the set of irreducible Brauer characters.

进一步，当正规子群为单位元群时，前面一个空间的基就是不可约Brauer特征标的集合。

For example, take the identity skeleton of a group of order 6 of the second type outlined above

例如，选取上述第二种类型的 6 阶群的单位元构架

However, a quasigroup which is isotopic to a group need not be a group.

但是，如果一个拟群与某个群同痕，由于缺乏单位元，拟群本身不一定是群。

The so-called AT-algebras are inductive limits of finite direct sums of matrices over the extension algeras of circle algebra by K, where K is the C~*— algebra of all compact operators on a separable infinite dimensional Hilbert space.

若V_*与V_*同构，且保持单位元等价类；T与T仿射同胚，且同构映射与同胚映射相容，则存在E与E′的同构导出上述同构和同胚，所谓AT-代数即为圆代数通过κ的本质酉扩张的矩阵代数的有限直和的归纳极限，这里κ为可分的无限维复Hilbert空间上的紧算子全体，不变量中的V*为三变元Abel半群，T为迹态空间，[1]为单位元所在的Murray-von Neumann等价类，r_E为连接映射。

The invariant V (A , [1〓] that we used for unital case is the semigroup of Murry-von Neumann equivalence classes of projections in matrices over C〓-algebras together with the class of the unit.

我们用来分类的不变量（V，[1〓]）是A的矩阵代数中所有投影的Murry-von Neumann等价类所成的半群及单位元所在的等价类。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失