Let G be a group. (a) >If |G| = 247 prove that every subgroup H G is cyclic.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: If H is a finite subgroup of a group G and x ∈ G, prove that |N(H)| = |N(xHx-1)|.

A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

A:

Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: Let H be a subgroup of a group G. Prove that the number of conjugates of H in G is |G:N(H)|.

A: It is given that H be a subgroup of a group G. It is required to prove that the number of conjugates…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

Q: If G is a group, prove that Aut(G) and Inn(G) are groups

A:

Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

A: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of…

Q: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

Q: Let G = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

A:

Q: Determine the order of (Z ⨁ Z)/&lt;(2, 2)&gt;. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: If G is a group of odd order and x ∈ G, show that x-1 is not in cl(x).

A: It is given that, G be a group of odd order and x∈G. Then, we have to show that x-1 is not in cl(x).…

Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

A:

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: Let H and K be subgroups of a group G and assume |G : H| = +0. Show that |K Kn H |G HI if and only…

A: Given:

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

A:

Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: Let G be a non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

A:

Q: Define X = R \ {k} and define to be the operation such that x+y= = (x − k)(y − k) + k. Check each of…

A: The given set is X=ℝ-{k} and the operation * is given by x*y=(x-k)(y-k)+k. (X,*) will be called a…