Question

this is under the study of group symmetry

So assume an element x in the group G has the order of infinity i.e. |x|=∞

find all generators of <x^{5}>

Answer #1

There are only two generators, ie x^5 and x^(-5) the inverse of x^5

Reason In an infinite cyclic group only two generators are there.

Proof

Let <a> be an infinite cyclic group Then |a| is infinity. Suppose b is an another generator then

<a> = <b> . Since b belongs to <a> we have b = a^n for some n

Then Since a belongs to <b> , a = b^m. This implies a = (a^n)^m = a^mn

ie, a = a^mn . Then mn =1 so there are two possibility

m =1 and n=1 or m=-1 and n=-1

Then b = a or b= a^-1

So there are only two generators . Here |x| is infinity . so |x^5| is also infinity

so <x^5> is an infinite cyclic group . So the generators are x^5 and x^(-5)

find all generators of Z. let "a" be a group element that has
infinite order. Find all the generators of . Please prove and
explain in detail please use definions and theorems. please i
reallly want to understand this.

Let X be an object that has exactly three symmetries, so its
symmetry group Sym(X) has exactly three elements. Show that X is
chiral.
Please show detailed prove!!! Thanks!

1. Let a and b be elements of a group, G, whose identity element
is denoted by e. Assume that a has order 7 and that a^(3)*b =
b*a^(3). Prove that a*b = b*a. Show all steps of proof.

suppose every element of a group G has order dividing 2. Show
that G is an abelian group.
There is another question on this, but I can't understand the
writing at all...

the question says:
prove that if a is an element of a group G,
then the order of a = order of its inverse.
my attempt:
Let order of a=n , so aⁿ=e , and so (a)ⁿ(a^-1)ⁿ=e=(a^-1)ⁿ , so
order of a divides order of a^-1
let order of a^-1 =m. so (a^-1)^m=e if and only if a^m =e , so
order of a^-1 divides order of a
so they are equal.
Q.E.D
is the proof correct?

Let G be a group and a be an element of G. Let φ:Z→G be a map
defined by φ(n) =a^{n} for all n∈Z. Find the image φ(Z) and prove
that φ(Z) a subgroup of G

2. Let a and b be elements of a group, G, whose identity element
is denoted by e. Prove that ab and ba have the same order. Show all
steps of proof.

4. Let f : G→H be a group homomorphism. Suppose a∈G is an
element of finite order n.
(a) Prove that f(a) has finite order k, where k is a divisor of
n.
(b) If f is an isomorphism, prove that k=n.

Let a be an element of order n in a group and d = gcd(n,k) where
k is a positive integer.
a) Prove that <a^k> = <a^d>
b) Prove that |a^k| = n/d
c) Use the parts you proved above to find all the cyclic
subgroups and their orders when |a| = 100.

Let G be a group
containing 6 elements a, b, c, d, e, and f. Under the group
operation called the multiplication, we know that ad=c, bd=f, and
f^2=bc=e. Which element is cf? How about af? Now find a^2. Justify
your answer.
Hint: Find the
identify first. Then figure out cb.

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