Let G be a finite group of order n and identity element 1. a. Show that g" =1 for any g e G. b. An element g € G is said to be square if there exists z e G such that g = x?. Prove that G has odd order if and only if every element of g is a square.

Q: Prove that a group of order 28 must have an element of order 2.

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Q: Let a and b be elements of a group G and H and K be subgroups ofG. If aH = bK, prove that H = K.

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Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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

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Q: 1. Let G be a p-group acting on a finite set X. Let XG be the set of fixed points of this action: XG…

A: The solution lies using the Orbit Stabilizer Theorem

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Q: (1).* 4 and |6| = 2, Let G be a group. Let a, b EG such that |a| and a b = ba. Then determine |ba|,…

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Q: Let H be a subgroup of G, and let a and b belong to G. Then |aH| = |bH|

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Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

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Q: Show that if E is not measurable, then there is an open set O containing E that has finite outer…

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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 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: 3. Let G be a group, HAG, KAG,HnK = {e}. Show that G= G/H x G/K.

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Q: c. Define the maps 71 : G x H → G and 72 : G × H → H by 11(9, h) = g and 72(9, h) = h respectively.…

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