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$\begingroup$Elements of $\math{R}/2 \math{Z}$ are of the form ex + 2\math{Z}$ for ex \in \math{R}$ and a complete set of representatives is given by choosing ex from $[0,2)$, although many other choices are possible. Elements of $\math{Z}/2 \math{Z}$ are of the form $ z + 2\math{Z}$ for $ z \in \math{Z}$ and a complete set of representatives is given by choosing $ z $ from $\{0,1\}$.

We may find all the elements of the equivalence class containing ex by \begin{align*} \big cup_{z \in \math{Z}/2\math{Z}} & x + 2\math{Z} + z \\ &= \left(x + 2\math{Z} + (0 + 2\math{Z}) \right) \cup \left(x + 2\math{Z} + (1 + 2\math{Z}) \right) \\ &= \left(x + 2\math{Z} \right) \cup \left(x + 1 + 2\math{Z} \right) \\ &= x + (2\math{Z} \cup (1 + 2\math{Z})) \\ &= x + \math{Z} \text{.} That is, the equivalence classes of the quotient defining AGE are the same as those in $\math{R}/\math{Z}$ (and addition goes through without complications).

$\begingroup$You're approaching the problem from the wrong side, in my opinion. The problem you're facing is due to how you identify $\math{Z}/2\math{Z}$ with a subgroup of $\math{R}/\math{Z}$ after applying the isomorphism.

Edit (Nov 9, 2014), suggested by Sam Head ... or is virtually cyclic, or is two-ended, or contains no Abelian free subgroups...

(I learned this from Danny Calegari, but I believe this is originally due to Groove.) The proof of this is quite slick and relies on the “Random Pigeonhole Principle”: That because of the number of pair of relaters are so large you are likely to get a number of almost equal relaters, that differ at a single letter, which kills off a generator.

I mean first the group of automorphism BC/RJ which consists of complex conjugation and identity. Several important subjects, like “real algebraic geometry” are based on these properties.

Once I was strongly tempted to call my paper “Some applications of representation theory of the group of 2 elements”, but my co-author convinced me that this title would be too radical. The paper I mentioned is: Wrong map and Grassmannians of real codimension 2 subspaces, Computational Methods and Function Theory, 1 (2001) 1-25.

$\begingroup$An example from differential geometry: If M is a compact, even-dimensional Riemannian manifold of positive sectional curvature, then its fundamental group is either 1 or Z / 2Z. Equivalently: a group acting freely on a compact, even-dimensional Riemannian manifold of positive sectional curvature is 1 or Z / 2Z.

If AGE is a group and all cyclic subgroups of AGE are of order 2, it immediately follows that AGE is Abelian. $\begingroup$RH holds if and only if the group of isometrics of the complex plane that preserve globally the multi set of non-trivial zeroes of the Riemann Zeta function is isomorphic to $ Z / 2Z $ (otherwise, it would be isomorphic to $ Z / 2Z \times Z / 2Z $).

$\begingroup$Edit after Emil's comment (so my answer is not wonderful then): It's a group (or any product of it) where addition and subtraction seen as a binary (that is, $- = + \CIRC (I'd, i)$ where in is the inverse map) actually coincides. This ultimately allows for the computation of the Poincaré polynomial of the real module spaces $\Mather{X}_r(SL(3,\math{R}))$.

$\in-group $\begingroup$Recently I gave a lecture to master's students about some nice properties of the group with two elements $\math{Z}/2\math{Z}$. Typically, I wanted to present simple, natural situations where the only group satisfying the given constraints is $\math{Z}/2\math{Z}$ (also $\math{Z}/2\math{Z}$ as a ring or as a field could qualify, but I'd prefer to stick to the group if possible).

Let AGE be a nontrivial quotient of the symmetric group on in>$4 letters (nontrivial meaning here different from 1 and the symmetric group itself). Let OK be an algebraically closed field and let OK_$0 be a subfield such that OK/k_$0 is finite.

This is a theorem of Emil Martin and I actually did not prove it because my students did not have enough background in field theory. Let OK be a field with the following property: there exists a $k$-vector space HE of finite dimension in>$1 and an isomorphism HE\some EX*$ between the space and its linear dual which does not depend on the choice of a basis, i.e. is invariant under $\text{GL}(E)$.

Then OK=\math{Z}/2\math{Z}$, in=$2 and the isomorphism HE\some EX*$ corresponds to the nondegenerate bilinear form given by the determinant. A homomorphism is a map of:R\to SO satisfying of(a+b)=f(a)+f(b)$ for all a, b \in RJ, and of(ab)=f(a)f(b)$ for all a, b \in RJ.

A objective ring homomorphism is called an isomorphism. Then we have a ring isomorphism \ Let us put of(2)=3a$ for some integer a.

Let IN be the subset of ARE defined by \ Then prove that IN is an ideal of the ring ARE. Moreover, show that IN is maximal and determine Characteristic of an Integral Domain is 0 or a Prime Number Let ARE be a commutative ring with $$1.

Show that if ARE is an integral domain, then the characteristic of ARE is either $$0 or a prime number up. Let \ Then the usual matrix addition and multiplication make ARE a ring.

Let \ Generators of the Augmentation Ideal in a Group Ring Let ARE be a commutative ring with $$1 and let AGE be a finite group with identity element he. Then the map $\epsilon: KG \to RJ defined by \ where a_i\in RJ and KG=\{g_i\}_{i=1}and, is a ring $(x^3-y^2)$ is a Prime Ideal in the Ring OR$, ARE is an Integral Domain.

Prove that the principal ideal $(x)$ generated by the element ex in the polynomial ring OR$ is a prime ideal if and only if ARE is an integral domain. Prove also that the ideal $(x)$ is a maximal ideal if and only if ARE is a If ARE is a Northern Ring and of:R\to R’$ is a Subjective Homomorphism, then OR’$ is Northern Suppose that of:R\to R'$ is a subjective ring homomorphism.

Actually I'm stupid today, it happens once in a while that I get extremely lazy and stupid in mathematics, but today I came up with a bizarre thing in abstract algebra that I couldn't find my mistake on my own, and I'm not sure whether what I've concluded is true or wrong, I was proving another theorem that one of my assumptions implied: (\BBB{2Z},+)\Cong (\BBB{4Z}, +). It's obvious that f is surjective, and it's easy to verify that f is invective because if f(m)=f(n) then 2 m=2n, and we obtain: m=n.

Anyway, is it true that every infinite cyclic group has a subgroup of it which is isomorphic to itself? Actually proving that every infinite cyclic group has such a group is a bit hard, I could prove it in an easier way. If G is cyclic, then all elements of G can be generated by a specific element like a. so we have G= IFF g = an n for any g in G. Now, every subgroup of G is cyclic too, this can be proved easily, therefore there must be a power of a that generates S (S being a subgroup of G).

So the only thing I need to do is to map a to an m where m is the power of a that generates the subgroup and everything is just like before in integers. The first part of this work established, with examples, the fact that there are more than one non-American isomorphic types of groups of order n = SP, (s, p) = 1, where s

The defining relations of such non-American isomorphic groups were outlined and a scheme developed to generate the numbers for the non-American isomorphic types of such groups. The scheme helped in generating many examples of non-American isomorphic types of such groups.

The situation where p k (mod s), k > 1 was worked out and such groups have no non-American isomorphic types. This gave credence to the fact that a group of order 15 and its like do not have a non-American isomorphic type.

Later, proofs of the number of non-American isomorphic types for n =SP and n =spy using the examples earlier generated were given. 12 2.1 ISOMORPHIC TYPES OF SMALL GROUPS OF ORDER n=SP 15 2.2 PROPOSITION 17 2.3 COROLLARY 18 2.4 GROUPS OF ORDER p2q 18 2.4.1 Proposition 21 2.4.2 Summary of Defining Relations 21 2.4.3 Remark 22 2.4.4 Proposition 23 2.4.5 Summary of Defining Relations 24 2.4.6 Proposition 25 2.4.7 Summary of Defining Relations 25 2.4.8 Proposition 27 2.4.9 Summary of Defining Relations 28 CHAPTER THREE METHODS AND GENERATION OF NON-ABELIAN ISOMORPHIC TYPES 3.0 NON-ABELIAN TYPES OF SOME GROUPS OF ORDER 2p.

It is due to this association that groups arise in different areas like Quantum Mechanics, Crystallography, Biology, and even Computer Science. There is no such easy definition of symmetry among objects without leading its way to the theory of groups.

Classifying groups arise when trying to distinguish the number of isomorphic groups of order n. In organic chemistry, conformal factors affect the structure of a molecule and its physical, chemical and biological properties. For instance, the position of atoms, relative to one another affects the structural formula of Hydrogen peroxide, H2O2.

According to Francis A Carey (2003) one could also write an infinite number of non-planar structures by tiny increments of rotation about the 0 – 0 bonds; Francis A Carey (2003). 1.1 STATEMENT OF THE PROBLEM Classifying groups arise when trying to distinguish the number of isomorphic types of a group of order n. Hall Jr and Senior (1964) used invariants as the number of elements of each order k (k small) to determine whether two groups of order 2n (n < 6) are isomorphic.

Philip (1988) in his article developed a systematic classification theory for groups of prime 14 power orders. Newman and O’Brien (1990) introduced an algorithm to determine up to isomorphism the groups of prime-power order.

It was introduced by Cayley who constructed the groups of order 4 and 6 in 1854. Determination of isomorphic types has been a comparatively difficult problem as there was no method that is sufficiently effective.

This is possibly due to the complexity of computation as the factors increase. The problem then arise to find the non-American isomorphic types of groups of higher orders which can be factorized into two or three distinct primes taking into consideration the relationship between the prime factors.

The need also arise to construct a suitable computer program to assist in solving such a problem. Hence, the statement of the problem is “Determination of the Number of non-Abelian Isomorphic Types of Certain Finite Groups”.

We hope to achieve the following objectives: (i) Finding relationship, through series of examples, of the number of non- Abelian Isomorphic types of groups of order n=SP and the congruence relation between the primes s and p. (ii) Determining the proof for the number of non-Abelian isomorphic types in each congruence relationship and stating their defining relations. (iii) Determine and design a suitable computer program that will help in working out the number relationship between such primes and generating the numbers for the non-Abelian isomorphic types.

(iv) Finding the non-Abelian isomorphic types of groups of order n = spy where s, p and q are distinct primes and determining their defining relations. The identity, rotation r through about its center, and the reflections 1 2 q, q in its two diagonals form a group under composition whose 16 multiplication is given in table 1 below.

Table 1.1: Four plane symmetries of a chessboard e r 1 q 2 q e e r 1 q 2 q r r e 2 q 1 q 1 q 1 q 2 q e r 2 q 2 q 1 q r e It is easy to check that multiplication modulo eight makes the numbers 1,3,5,7 into a group. There is an apparent similarity between these two groups if we ignore their origins.

Isomorphism also preserves the order of each element (Armstrong, 1988). (ii) Every cyclic group of infinite order is isomorphic to the additive group I of integers Proof: Consider the infinite cyclic group G generated by a and the mapping n an, no I of I into G. Now, this mapping is onto since any n in I is mapped to exactly one an.

Hence, the mapping is an isomorphism, that is I G. (iii) The group Z of integers (with addition) is a subgroup of R, and the factor group R/ Z is isomorphic to the group S’ of complex numbers of absolute value 1 (with multiplication): 18 R/ Z S’ An isomorphism is given by f(x+ Z) = e2xi for every x in R. Proof: We only need to show that for any k Z, then f(x + k) = e2i(x +k) = e2xi +2ik = e2xi. Also, for z R, then f(LN(z + k)) = ELN(z + k) = z + k. (iv) The Klein four-group is isomorphic to the direct product of two copies of Z2 = Z / 2Z and can therefore be written Z2xZ2.

(v) Generalizing this, for all odd n, D2n is isomorphic with the direct product of DN and Z2. The order of a group G, written G is the cardinal number of elements of G. G is said to be finite or infinite according as its order is finite or infinite (UK, 1980).

The smallest subgroup generated by an ANB b is denoted by a, b . m ms A C C C Where i m 11 for all i= 1,2,…,s-1 1.4.14 THEOREM If H and K are normal subgroups of G such that HK 1 then any element x, of CHG commutes with any other element y of K. PROOF: For any mph and yak, consider the commentator 21 1 1 1 1 z LyX y LyX y x ex y and notice from the first factorization and the normality of K that y-1K, by, ex, yGG -1 OK z OK.

PROOF: Suppose G/K OK, TK, …, t KM n1 Then for any x, YAG we have x=, you, Gy t v, s r For some u, vK and thus by, y= Gustav = TK+run = try+SVP = trots = y x, YAG (Since u, v permute with t). This contradicts the non-Abelian hypothesis on G. The problem of explicitly constructing all the groups of a given finite order has a long and somewhat conquered history; its study was initiated by Cayley in 1864 when he determined the groups of order at most 6.

The aim is to determine a complete and 22 redundant list of the groups of a given order: a representative of each isomorphism type present. It is usually comparatively easy to generate a complete list; the difficulty lies in the reduction to distinct isomorphism types (Hall M., 1976).

Thus, in this case, there is an element of order p which permutes with every element of K. 1.4.18 POLYNOMIAL: A function of z of the form 23 P(z) = a0 + a1z + a2z2 + … + anon, in which an 0 is called a polynomial of degree n in z.