If G is a finite group with |G|<120 and G has subgroups of orders 10, 15 and 20 then the order of G is: * O 30 60 80 O 90

Q: Let G be the symmetry group of a circle. Explain why G containsDn for all n.

A: Given: G is the symmetry group of a circle.We need to explain that G contains Dn for all n.

Q: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Given : G is a finite simple group and contains subgroups H and K such that G:H and G:K are prime.…

Q: 2. Let G be a group. Recall we showed in class that if |G| is finite, then o(g) is finite for all g…

A: be a group.If  is finite then  is also finite where We have to show the converse of the statement…

Q: Let H1 and H2 be finite groups, and let K = H1 × H2 be the product group. Let C1 be a conjugacy…

A: Given: H1 &amp; H2 are finite groups. K=H1×H2 is a product group. C1 is a conjugacy class of H1…

Q: Suppose that G is a finite group and G has exactly one subgroup foreach divisor of |G|. Prove that G…

A: Given: G is a finite group and has exactly one subgroup for each divisor of G.We need to prove that…

Q: If G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

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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 Gbe a finite group and let H be a p-sylow group of G. If Nis a normal sub group of e prove that…

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Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

Q: Let G be a group and let x ∈ G. Prove that if a has order of 12, then x3 has order 4.

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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 and g E G, show that the number of conjugates of g E G is [G : CG(g)]

A: Given G be a group and g∈G be an element. Let Bg be the set of all conjugate elements of g∈G.…

Q: Every subgroup of a group G is normal * False True

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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: Let G be a group which is an algebraic structure.Let H and K be the subgroups of G with H⊂K⊂G. Let…

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Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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Q: 1 c = |G| gEG

A: The centralizer of a group G is the set of all elements in G that commutes with a particular element…

Q: Let G be a group with p or more distinct elements of order p for some prime p. Show that G is not…

A: Given : Let G be a group which has p  elements of order p, p is a prime no.  To show G is not cyclic…

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

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Q: If G is a finite group and a ∈ G, prove that |a| is finite. Hint: You just need to show that there…

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Q: Prove Corollary 36.4 which says, "Let G be a finite group. Then G is a p-group if and only if |G| is…

A: First suppose that G is a p-group. Let q is a prime which divides |G|, then G contains an element of…

Q: 3. Prove or disprove: If G is a group, (g¬')-' = g.

A: Consider the given information: Let G is a group. To show that (g-1)-1=g