- Check the order of the group. For example, if it is 15, the subgroups can only be of order 1,3,5,15.
- Then find the cyclic groups.
- Then find the non-cyclic groups.

Contents

- How do you find cyclic subgroups?
- Can a non cyclic group have a cyclic subgroup?
- What is a nontrivial cyclic group?
- Which of the following group is non cyclic?
- Are all subgroups cyclic?
- How do i find all the subgroups of a given group?
- How do you find the number of subgroups?
- What is the order of smallest non cyclic group?
- How do you prove a subgroup is cyclic?
- How do you find a trivial subgroup?
- What is a proper non trivial subgroup?
- How do you show a group is trivial?
- Is Zn always cyclic?
- Which of the following group is non cyclic group but all of its subgroups are cyclic?
- Is Z5 cyclic?

For a∈G we call ⟨a⟩ the cyclic subgroup generated by a. **If G contains any element a such that G=⟨a⟩, then G is a cyclic group** . In this case, a is a generator of G. If a is an element of a group G, we define the order of a as the smallest positive integer n such that an=e, and we write |a|=n.

Therefore we have proved the following theorem: **Every non-cyclic group contains at least three cyclic subgroups of some order**. arbitrary proper divisor of the order of the group. since G is not cyclic and thus it is proved that g is not divisible by more than two distinct primes.

Explanation. **If a nontrivial group has no proper nontrivial subgroup, then it is a cyclic group of prime order**. In other words, it is generated from a single element whose order is prime.

∴**{1,3,5,7} under multiplication mod 8** is not a cyclic group.

**All subgroups and quotient groups of cyclic groups are cyclic**. In particular, all subgroups of Z are of the form ⟨m⟩ = mZ, where m is a positive integer.

The easiest way to find subgroups is to **take a subset of the elements and then find all the power products of those elements**. Suppose you have two elements a, b in your group, then you need to consider all strings of a, b which gives 1, a, b, a2, ab, ba, b2, a3, aba, ba2, a2b, ab2 , bab,b3,…

To determine the number of subgroups of a given order in an abelian group, one needs to know **more than the order of the group**, since there are, for example, two distinct groups of order 4 , and one of them has one subgroup of order 2, the other has order 3.

The little four group is the smallest noncyclic group. However, it is an abelian group and isomorphic to the dihedral group of **order (cardinality) 4**, i.e. D_{4} (or D_{2}, using die geometric convention); apart from the group of order 2, it is the only dihedron group that is abelian.

Theorem: All subgroups of a cyclic group are cyclic. If G=⟨a⟩ is cyclic, then for every divisor d of |G| there is exactly one subgroup of order d that is defined by a|G|/d a | can be generated G | / i . Proof: Let |G|=dn | G | = d n .

A subgroup N of a group G is properly called if N≠G, and not trivially, **if N≠{e}, where e is the identity of G**. For example, N={0,2} is a proper subgroup of (Z/4Z,+), isomorphic to Z/2Z.

**The quotient group of any group G itself is the trivial group**: G/G=1, and the quotient projection G→G/G=1 is the unique such group homomorphism. It cannot be trivial to use a group presentation to decide whether a group so presented is trivial, and in fact the general problem is undecidable.

**Zn is cyclic**. It is generated by 1. Example 9.3. The subgroup of 1I,R,R2l of the symmetry group of the triangle is cyclic.

The group **U(8) = 11,3,5,7l** is non-cyclic since 11 = 32 = 52 = 72 = 1 (so there are no generators). The only correct subgroups are 11l, 11.3l, 11.5l and 11.7l, all of which are obviously cyclic.

The group (Z5 × Z5, +) is **non-cyclic**.

- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/04%3A_Cyclic_Groups/4.01%3A_Cyclic_Subgroups
- https://www.jstor.org/stable/85619
- https://groupprops.subwiki.org/wiki/No_proper_nontrivial_subgroup_implies_cyclic_of_prime_order
- https://www.competoid.com/quiz_answers/14-0-64247/Question_answers/
- https://en.wikipedia.org/wiki/Cyclic_group
- https://math.stackexchange.com/questions/2082441/how-do-i-find-all-all-the-subgroups-of-a-group
- https://math.stackexchange.com/questions/205918/determining-number-of-subgroups
- https://en.wikipedia.org/wiki/Klein_four-group
- https://crypto.stanford.edu/pbc/notes/group/cyclic.html
- https://www.youtube.com/watch?v=wnIoupqmssw
- https://math.stackexchange.com/questions/331631/what-does-it-mean-to-have-no-proper-non-trivial-subgroup
- https://ncatlab.org/nlab/show/trivial+group
- https://www.math.purdue.edu/~arapura/preprints/algebra9.pdf
- https://faculty.weber.edu/mattondrus/4110/docs/chap4-HW.pdf
- http://www.math.lsa.umich.edu/~kesmith/Lagrange’sTheoremANSWERS.pdf

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