# Are cyclic groups subgroups?

## Are cyclic groups subgroups?

Theorem: All subgroups of a cyclic group are cyclic. If G=⟨a⟩ is cyclic, then for every divisor d of |G| there exists exactly one subgroup of order d which may be generated by a|G|/d a | G | / d .

## What do you mean by cyclic group and subgroup?

Definition and notation For any element g in any group G, one can form a subgroup of all integer powers ⟨g⟩ = {gk | k ∈ Z}, called a cyclic subgroup of g. A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator.

How do you determine the number of Homomorphisms between two groups?

For finding homomorphism f for arbitraay two groups, use the following facts:

1. |f(g)| divides |g| where g belong to the domain with |g|<∞ [this is useful for finite groups]
2. f(gn)=[f(g)]n.
3. List all normal subgroups of domain and use first isomorphism theorem.

### Is there always a Homomorphism between groups?

There’s always a homomorphism between any two groups — the trivial one (all elements of the domain are mapped to the identity element of the codomain group).

### How many subgroups are in a cyclic group?

A finite cyclic group has exactly one subgroup for each divisor of the order, so if the order is 6, that makes 4 subgroups.

Do cyclic groups have proper subgroups?

A cyclic group G has only one proper subgroup and it is of order p (a prime).

#### How many group homomorphisms are there from?

There are four such homomorphisms. The image of any such homomorphism can have order 1, 2 or 4. If it has order 1, then φ maps everything to the identity or φ(x) = (0,0.

#### How many group homomorphisms are there from S3 to a4?

There are 34 homomorphisms from S3 to S4. Let’s counting homomorphisms by analysis of its kernel.

Do homomorphisms preserve identity?

A direct application of Homomorphism to Group Preserves Identity.

## How many cyclic subgroups does Z 16 have?

one subgroup
Z16: A cyclic group has a unique subgroup of order dividing the order of the group. Thus, Z16 has one subgroup of order 2, namely 〈8〉, which gives the only element of order 2, namely 8. There is one subgroup of order 4, namely 〈4〉, and this subgroup has 2 generators, each of order 4. 