2) Let G = V4.Show which of the following subsets of G is a subgroup of it? a) H = {e, a} b) K = {e, b} c) L = {e,c} d) M = {e,b,c} 3) Let G = GL(2, R) and H = {[a b], ad #0}. Prove that H ≤ G? 4) Let G = (Q*,.) and H = {2", n E Z}. Prove that H ≤ G? 5) Try to find all subgroups of all groups we are study in lecture 1 and 2

Q: 9. Which of the following groups are cyclic? Justify your answer. a) Z;×Z12 b) Z1,×Z, c) Z2×Z8×Z16…

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Q: Exercise 3. Let T be a set such that |T| ≥ 3 and let G = Sym(T) be the group of permutations of T.…

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Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

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Q: ) If we have an epimorphism from G to G’, then we know G must be the same or of bigger order than…

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Q: Let H = {a + b√3 | a, b ∈ Q and a and b are not both zero}. Prove or disprove: H is a subgroup of…

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Q: Let C2 = {±1} be a subgroup of R". Prove that R = R+ x C2 where R is the set of real numbers.

A: Please check step 2 and 3 for the solution.!

Q: Let G be group. Assume that Set H₁ = and H₂ =<b) Assume \ H₁|-21 and | H₂ | =10 a) Show that HinH₂…

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Q: 2. Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?…

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Q: Let Z be the group of the integes. oddition. Let f:232120e tre homonorphism fiam 2oZg. defined with…

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Q: 2. Describe the following sets: 47 η 6Z = Ζ: π ε 47 and πε 6Z)

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: SIZ): Set of all fiZ-→Z which are both one-one S(Z): on ti and S(Z) is a group under composition of…

A: Given, S(ℤ)=f|f:ℤ→ℤ, 1-1, onto is a group under composition of function. And H=fn:ℤ→ℤ|fn(x)=n+x.…

Q: Q,+ ,+,*) is not a proper subgroup of R+ under multiplication? T or F

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Q: Show that an isometry of Rn is one-to-one.

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Q: i) Show that the set of translations in space (in two or three dimensions) forms a group for a given…

A: As per policy at a time one question or a question with related subparts can be answered. We will…

Q: {[::] 1) 5. Assume that R a, b, c, d e Z and R' [a] (b) a, b, c, d e Z2 Given that %3D 0: R → R…

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Q: let by a fi

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Q: . Prove that the shift o is a homeomorphism.

A: To, prove that σ is homomorphism we will show that the inverse image of open set is open. So, the…

Q: Show that an isometry of Rn is one-to-one.

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: ► Exercise 11.2. Prove Proposition (c) A finite product of covering maps is a covering map.

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Q: n. Is it like cosets or something，a clear

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Q: Subgroup of a sets of lefL is b) suppose thkat a Group G and and A,B are the right coseets of H in G…

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