What is meant by abelian group?

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

What do you mean by solvable group?

A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices are all prime numbers. Solvable groups are sometimes called “soluble groups,” a turn of phrase that is a source of possible amusement to chemists.

What is cyclic group math?

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.

What do you mean by permutation group?

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The term permutation group thus means a subgroup of the symmetric group.

Which is the smallest Abelian group?

The smallest noncyclic group is the four element Klein four-group https://en.wikipedia.org/wiki/Klein_four-group . All finite abelian groups are products of cyclic groups. If the factors have orders that are not relatively prime the result won’t be cyclic.

How do you classify Abelian groups?

Abelian groups can be classified by their order (the number of elements in the group) as the direct sum of cyclic groups. More specifically, Kronecker’s decomposition theorem. An abelian group of order n n n can be written in the form Z k 1 ⊕ Z k 2 ⊕ …

Are dihedral groups solvable?

All of the dihedral groups D2n are solvable groups. If G is a power of a prime p, then G is a solvable group.

Are all cyclic groups Abelian?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.

Why is it called the symmetric group?

For example, the symmetry group of an icosahedron? Think of it as permutations of the vertices. i.e. every “symmetry group” is a subgroup of this universal group. So it’s called the “symmetric group” because of that.

What do you mean by simple group?

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups include the infinite families of alternating groups of degree. , cyclic groups of prime order, Lie-type groups, and the 26 sporadic groups.

What can a silyl group be used for?

Silyl groups can conveniently be used as protection for an imine nitrogen. However, the most convenient method for the preparation of trimethylsilylimines is the reaction of nonenolizable aldehydes with lithium hexamethyldisilazane.

What is the definition of the radical silyl?

Definition of silyl. : the univalent radical SiH3 derived from monosilane by removal of one hydrogen atom.

Where is the silyl group located in Birch reduction reaction?

M.C. Elliott, N.L. Hewitt, in Comprehensive Organic Synthesis II (Second Edition), 2014 Silyl groups are generally tolerated in Birch reduction reactions. In such cases, the silyl group in the product will be located on a sp3 -hybridized carbon atom in the absence of other directing effects.

Which is the best definition of silyl gas?

1. a gas with an unpleasant odor, SiH 4, soluble in water: used as a dopant for semiconductors in the production of solid-state devices. 2. any of a class of silicon hydrides analogous to the alkanes. [< German Silan (1916); see silicon, -ane]