**ORDERS OF ELEMENTS IN A GROUP Introduction UCONN**

Theorem: All subgroups of a cyclic group are cyclic. If \(G = \langle g\rangle\) is a cyclic group of order \(n\) then for each divisor \(d\) of \(n\) there exists exactly one subgroup of order \(d\) and it can be generated by \(a^{n/d}\).... How do I check if an element a belongs to a specific cyclic group G of prime order, given the generator? Right now i simply generate all the elements in the group, save …

**Groups of order 8 Groupprops**

n=8: Now let G be a group of order 8. If G has an element of order 8, then G is cyclic, so G is isomorphic to Z=8Z: So assume that G has no element of order 8; by Lagrange, every non-... x11 Direct Products and Finitely Generated Abelian Groups Homework: 3-7, 9, 10, 15-18, 21-24, 46, 52 3. Find the order of (2;6) in Z 4 Z 12. (2;4) = 2 and 4

**How to find generator $g$ in a cyclic group? Stack Exchange**

Instructions for Exercises 2-4: For a cyclic subgroup of order 4, list the elements of the subgroup, identify at least one generator for the subgroup, and show how …... { Seeking a contradiction, let G be a group of order 3 that is not cyclic. Thus G has an identity Thus G has an identity element e, and two additional elements, call them aand b.

**Finding the order of a cyclic group? Yahoo Answers**

31/03/2012 · This group has 8 proper subgroups, 1 cyclic of order 4, 5 cyclic of order 2, (you've listed these, along with the trivial group {d}, and 2 isomorphic to Z2 x Z2. Those are the two you are missing, and are the hardest to see (at least for me). They arise from the 180 rotation and either a flip on a diagonal or a flip on the vertical or horizontal.... Show that any cyclic group of even order has exactly one element of order 2. Solution: If G is cyclic of order 2n, for some positive integer n, then it follows from Theorem 3.5.2 that G is isomorphic to Z 2n .

## How To Find The Order Of A Cyclic Group

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## How To Find The Order Of A Cyclic Group

### 2.16 Let Gbe a cyclic group of order n. Then an element x? Ggenerates Gif and only if #x= n. Then an element x? Ggenerates Gif and only if #x= n. Now ?xing a generator x? G, we have G= {e,x,x 2 ,...,x n?1 }, and so in view of the formula

- 4.2.2 Order of a Cyclic Group and of its Elements, Gen- erators of a Cyclic Group The next theorem and corollaries are related to the order of a k and the groups
- Order (group theory) 2 The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem).
- The group Z 4 is also cyclic of order 4. By theorem: "Any two cyclic groups of the same order are isomorphic.", it follows that h6i=h24iis isomorphic to Z 4. 5. S10MTH 3175 Group Theory (Prof.Todorov) Quiz 5 (Practice-Some Solutions) Name: 10. Let h8ibe the subgroup of Z 48. (a) What is the order of the factor group Z 48=<8 >?
- 30/03/2011 · Best Answer: The number of generators for a cyclic group of order n is ?(n). ?(n) denotes Euler's phi function, which counts the number of positive integers up to n which are relatively prime to n.

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