Challenge Yourself #CHAPTER2

If G is a Prime order group, then G has

No Proper Subgroup

No Improper Subgroup

Two Improper Subgroups

Let G be a group order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?

G is always cyclic, but H may not be cyclic.

G may not be cyclic, but H is always cyclic.

Both G and H are always cyclic.

Both G and H may not be cyclic.

If (G, ⋅) is a group such that (ab)-1 = a-1 b-1, ∀ a, b ∈ G, then G is a/an

Commutative semi group

Abelian group

Non-Abelian group

The number of generators of the cyclic group G of order 8 is

Any group of order 3 is

Cyclic and Abelian

Cyclic but not Abelian

Infinite cyclic group

A subgroup has the properties of ________.

Closure, associative

Commutative, associative, closure

Inverse, identity, associative

Closure, associative, Identity, Inverse

A trivial subgroup consists of ___________.

Identity element

Coset

Inverse element

Ring

A normal subgroup is ____________.

A subgroup under multiplication by the elements of the group.

An invariant under closure by the elements of that group.

A monoid with same number of elements of the original group.

an invariant equipped with conjugation by the elements of original group.

If a * b = a such that a ∗ (b ∗ c) = a ∗ b = a and (a * b) * c = a * b = a then ________.

* is associative

* is commutative

* is closure

* is abelian

The set of rational numbers form an abelian group under _________.

Association

Closure

Multiplication

Addition

If x * y = x + y + xy then (G, *) is _____________.

Abelian group

Commutative semigroup

Cyclic group

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