There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two elements Select one: True False

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: Let H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: 4. Let H & K are two subgroups or a group G such that H is normal in G then show that HK is a…

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Q: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…

A: We know that  b∈bH                    (1) We know that  aH = bH if and only if a-1b ∈H…

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: Given: 2Z is a subgroup of (Z,+). We have to find the right coset of -5+2Z.

Q: Let G be a group. Show that Z(G) = nEC(a). [This means the intersection of all subgroups of the form…

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Q: (b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.

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Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: Let H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

Q: 4. (a) Show that every group of order 4 is isomorphic to either Z4 or V4. (b) Show that H {1,…

A: We have show given property:

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: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

Q: Prove If S1 and S2 are subgroups of G, then S1 intersection  S2 is a subgroup of G.

A: Let S1 and S2 are two subgroups Then if x, y E S1 or S2 .xy E S1 or S2 And V x E S1 or S2 Then x-1 E…

Q: Q2)) prove that the center of a group (G, ) is a subgroup of G and find the cent(H) where H = (0, 3,…

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Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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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…

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Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H  and…

Q: Let H be a subgroup of a group G and a, b EG. Then b E aH if and only if O None of these O ab EH О…

A: The solution is :

Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…

A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…

Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and…

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Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

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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…

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Q: Let H be a subgroup of a group G and a, b € G. Then b E aH if and only it O None of these O ab e H O…

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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:…

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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…

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

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Q: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

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Q: Let H and K be two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

A: F hv

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba-1 E H ba E H O b-1a…

A: Q9. Third option is correct.

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

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Q: The order of the element (1,1) + ((4,4) in the factor group Z, xZ,/((4.4) is 4. Select one: O True O…

A: let G=G=ℤ6×ℤ8

Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

A:

Q: The group U(15) is an internal direct product of the cyclic subgroups generated by 7 by 11, U(15) =…

A: We  have to check that U(15)= &lt;7&gt;×&lt;11&gt;  Or not. Concept: If  n =p1.p2 Where p1 and p2…

Q: dicfin et Prove that a group G has exactly 3 6. - subgroups iff G is a ylic grop ef ender på pis…

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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

Q: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

Q: Let G be a group having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

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

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

A: Use the fact that order of subgroup divides order of group

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

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

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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…

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Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: If G is a finite group with |G|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

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Q: Let H be a subgroup of G and let a, be G. If Ha Hb, then* %3D aH = bH O a-1H = b-1H O Ha = Hb Ha-1 =…

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Q: Consider the group D4 and the subgroup {I, F}. List all the left cosets of H (with all the elements…

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Q: Let H be a subgroup of a group G and a, bEG. ThenbE aH if and only if * O None of these O ab EH O…

A: