# Is every abelian group is cyclic or not and why?

every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.

### Is every finite abelian group is cyclic?

No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic

### Is every cyclic group abelian?

Yes. Lets call the generator of the group z, then every element of the group can be written as zk for some k. Then the product of two elements is: zkzm=zk+m Notice though that then zmzk=zm+k=zk+m=zkzm, so the group is indeed abelian.

### Prove that a group of order three is abelian?

By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.

### Every subgroup of a cyclic group is cyclic?

Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.

### Prove that every a cyclic group is an abelia?

Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a

### Is the symmetry group of the square an abelian group?

Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not.

### What is the definition of an abelian group?

An abelian group is a group in which ab = ba for all members a and b of the group.

### What is an Abelian group?

a group or set of group is said to be abelian if the law of commutation is always held. it means if for a group or set S having elements a,b belongs to S then a*b=b*a then the group is called abelian group.

### Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?

No! Take the quaternion group Q_8.

### Is a group of order 24 abelian group or not?

The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24

### What does the term abelian mean?

The term abelian is most commonly encountered in group theory, where it refers to a specific type of group known as an abelian group. An abelian group, simply put, is a commutative group, meaning that when the group operation is applied to two elements of the group, the order of the elements doesn't matter. For example: Let G be a group with multiplication * or addition +. If, for any two elements a, b Є… Read More

### Prove that a group of order 5 must be cyclic?

There's a theorem to the effect that every group of prime order is cyclic. Since 5 is prime, the assertion in the question follows from the said theorem.

### A cyclic group of length 2 is called identity?

A cyclic group of order two looks like this. It has two elements e and x such that ex = xe = x and e2 = x2 = e. So it is clear how it relates to the identity. In a cyclic group of order 2, every element is its own inverse.

### Is every group whose order is less than or equal to 4 a cyclic group?

Yes. The only group of order 1 is the trivial group containing only the identity element. All groups of orders 2 or 3 are cyclic since 2 and 3 are both prime numbers. Therefore, any group of order less than or equal to four must be a cyclic group.

### What is abelianization?

Abelianization is a homomorphism which transforms a group into an Abelian group.

### Is it true that an infinite cyclic group may have 3 distinct generators?

A cyclic group, by definition, has only one generator. An example of an infinite cyclic group is the integers with addition. This group is generated by 1.

### What is the order of the cyclic group mean?

The order of a cyclic group is the number of distinct elements in the group. It is also the smallest power, k, such that xk = i for all elements x in the group (i is the identity).

### What is the number of groups of order 8 upto isomorphisms?

There are 5 groups of order 8 up to isomorphism. 3 abelian ones (C8, C4xC2, C2xC2xC2) and 2 non-abelian ones (dihedral group D8 and quaternion group Q)

### Pq plus qp?

If you are working with real numbers, or even complex numbers, pq is the same as qp, so the result is the same as 2pq. If you use some multiplication that is NOT commutative (such as, when you multiply matrices), you can't simplify the expression.

### Is every square cyclic?

Yes. The sum of opposite angles is 180 degrees and that is a necessary and sufficient condition for a quadrilateral to be cyclic.

### What is finite and infinite cyclic group?

Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g. If the operation is multiplicative then the elements are g0, g1, g2, ... Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such… Read More

### Let G be a cyclic group of order 8 then how many of the elements of G are generators of this group?

Four of them.

### How do you determine number of isomorphic groups of order 10?

There are two: the cyclic group (C10) and the dihedral group (D10).

### Is there a non-cyclic group with precisely four subgroups?

Yes. The group {0,1} has precisely 4 subgroups.

### What has the author Joel Samuel Georges written?

Joel Samuel Georges has written: 'Associativity conditions for division algebras corresponding to any Abelian group' -- subject(s): Abelian groups, Universal Algebra 'Introductory mathematical analysis' -- subject(s): Calculus, Functions

### What has the author Paul Edwin Lewis written?

Paul Edwin Lewis has written: 'Characters of Abelian groups' -- subject(s): Group theory

### What is a menstructual cycle?

Every women has a menstructual cycle. It is nothing but a cyclic discharge of every 33 to 39 days

### What is the difference between Linear Block Codes and Cyclic Codes?

linear codes and cyclic codes sub class of block codes where linear codes satisfies linearity property i.e. addition of any two code vectors produces another valid code vector where as cyclic codes satisfies cyclic shift property i.e. for every cyclic shift of a code vector produces another valid code vector

### What is Abelian algebra?

Abelian algebra is a form of algebra in which the multiplication within an expression is commutative.

### Would glucose solution produce a positive Benedicts test?

Yes. Glucose has an aldehyde group when it's not in it's cyclic form. considering there's a equillibrium between the chain and cyclic form, then the test = positive.

### What is a cyclic hydrocarbon?

It is a ring formed carbon skeleton. The smallest possible cyclic hydrocarbon is cyclopropane (triangular C3H6) together with cyclobutane both being rather unstable (degration to linear propene and 1-butene). Five and six membered rings are quite common. The most 'famous' cyclic hydrocarbon is the aromatics-group of benzene, which is the cyclic 'honeycomb' formed 1,3,5-cyclohexatriene C6H6 (and derivatives).

### Does a trapezium have rotational symmetry?

rectangle has inversion (180 deg rotation) hexagon has 60 deg ratation, cyclic group genterated is 60, 120, 180, 240, 300, 360=0 equilateral triangle has 120 deg rotation, cyclic group genterated is 120, 240, 360=0

### What is another name for commutative property?

The Abelian property, after the Norwegian mathematician, Niels Abel (early 19th Century) who made major contributions to Group Theory.

### How do we prove that a finite group G of order p prime is cyclic using Lagrange?

Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of… Read More

### How do you use cyclic in a sentence?

"The water cycle is a cyclic process is a cyclic sentence."

### Whats the difference between cyclic behavior and periodic behavior?

Cyclic behavior occurs in regular cycles ever year like, circadian rhythms and migration. Periodic behavior is cycles of activity that are synchronized with periodic events in the environment. The difference is cyclic behavior is a regular cycle every year and periodic behavior may change with the environment.

### Why sucrose non reducing sugar?

Sucrose is the combination of cyclic structures of Glucose and Fructose and therefore does not have a free aldehyde or ketone group.

### In group theory what is a group generator?

In abstract algebra, a generating set of a group G is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. More generally, if S is a subset of a group G, then <S>, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S… Read More

### What are the symbols of permutation groups?

some examples of symbols for permuation groups are: Sn Cn An These are the symmetric group, the cyclic group and the alternating group of order n. (Alternating group is order n!/2, n>2) One other is the Dihedral group Dn of order 2n.

### What has the author A B Katok written?

A. B. Katok has written: 'Lectures on surfaces' -- subject(s): Surfaces 'Rigidity in higher rank Abelian group actions' -- subject(s): Rigidity (Geometry), Abelian groups 'Invariant manifolds, entropy, and billiards' -- subject(s): Entropy, Global analysis (Mathematics), Invariant manifolds, Ergodic theory, Differentiable dynamical systems

### Is meiosis cyclic or non cyclic?

Meiosis is not cyclic; rather it is a linear process. It does not cycle.

### How is glucose created?

by the result of covalent bond between the Aldehyde C atom and the C-S hydroxy group to form Cyclic hemiactol.

### Do matrices form an abelian group under multiplication?

More precisely, I think you're asking whether the set of n X n matrices forms an abelian group under multiplication. The answer is no (assuming n>1). For example (1 0)(0 1) = (0 1) (0 0)(0 0) (0 0), but (0 1)(1 0) = (0 0) (0 0)(0 0) (0 0). However, the set of n x n diagonal matrices does form an Abelian set. This is true regardless of the direction of the diagonality… Read More

### What is required to excite the electrons of the chlorophyll and start the cyclic and non-cyclic phosphorylation?

required to excite the electrons of the chlorophyll and start the cyclic and non-cyclic phosphorylation?