3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and b. a E H and b E H imply ab e H.

Q: Show that if H and K are subgroups of G then so is H ∩ K.

A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…

Q: G. Show that if H and K are subgroups of an abelian group G, then {hk\h E H andke K} is a subgroup…

A:

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False

A: let H be a cyclic subgroup of a group G.

Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

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: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

A:

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 Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

A:

Q: Suppose H and K be subgroups of a finite group G with |G : H| = m and |G : K| = n. Prove that…

A: We use here, Tower law of subgroup which states that Let (G,∘) be a group.  Let H be a subgroup of G…

Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: It is not possible that, for a group G and H and K are nomal subgroups of G, H is isomorphic to K…

A: Let G be a group and H and K are normal subgroups of G

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

A:

Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

A:

Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: If H and K are two subgroups of finite indices in G, then show that H ∩ K is also of finite index in…

A: If H and K are two subgroups of finite indices in G, then show that H ∩ K isalso of finite index in…

Q: Let H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

A:

Q: 1- Prove that if (Q -(0),) is a group, and H = an, m e Z} 1+2m is a subset of Q - {0)}, then prove…

A: A subset H of a group G, · is said to be a subgroup of G, · if for any a,b∈H we have: a·b∈H a-1∈H…

Q: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

A:

Q: Prove or disprove, as appropriate: If G x H is a cyclic group then G and H are cyclic groups.

A: GIven two groups G and H such that GxH is cyclic. True or false: G and H themselves are cyclic

Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

Q: Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.…

A:

Q: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

A:

Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

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

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

A:

Q: 1.19. If M and N are normal subgroups of a group G, then G/(MN) is isomorphic to a subgroup of the…

A: Given :  M and N are normal subgroups of a group G. To prove :  G/M∩N is isomorphic to  G/M × G/N.

Q: Show that if G is a finite group with identity e and with an even number of elements, then there is…

A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…

Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

A:

Q: Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X…

A: Solution: Let us consider (G, .) be a group and B be a subgroup of the group G. Now for any g∈G, the…

Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

A:

Q: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

A:

Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

A:

Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A:

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

A:

Q: 64

A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…

Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

A:

Q: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

A:

Q: 7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).

A: 7) Let G be a group of order 1045 and H∈Syl19(G) , K∈Syl11(G).  To show: K⊲G and H⊆Z(G). As per…

Q: Show that if H and K are subgroups of an abelian group G then {hk: h element of H and k element of K…

A:

Q: 2. If H and K are subgroups of an abelian group G, then HK = {hk | h e H and k e K} is a subgroup of…

A: