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Some group $(G, o)$ is known to be abelian. Then, which one of the following is true for $G$?

  1. $g=g^{-1} \text{ for every } g \in G$

  2. $g=g^2 \text{ for every }g  \in G$

  3. $(goh)^2 = g^2oh^2 \text{ for every } g, h \in G$

  4. $G$ is of finite order

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Associativity property of Group.

  • For all $a, b$ and $c$ in $G$, the equation $(a o b) o c = a o (b o c)$ holds.

For an Abelian group, commutative property also holds. 

  • For all $a, b$ in $G$, the equation $a o b = b o a$

From option C, using these two properties,

$(goh)^2  = (goh) o (goh) \\= (h o g) o (g o h)  \\= ((h o g) o g ) o h \\= (h o (g o g)) o h \\= (h o g^2) o h \\= (g^2 o h) o h \\= g^2 o (h o h) \\= g^2 o h^2$

So, C is correct.   

Integer addition $(Z, +)$ is an Abelian group. 

Inverse of $1$ is $-1$ and not $1.$ So, A is false. 

$1^2 = 1 + 1 = 2 \neq 1$. So B also false. 

The order of a group is the number of elements in it. An integer is an infinite set, so D is also false. 
References:

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Some common Abelian groups:-

  • (Z, +)
  • (R, +)
  • (Ev, +) //Set of Even numbers on addition
  • (M, +) //Matrix addition

For addition, identity element, e = 0.

Inverse would be anything that results in 0 after addition.

 

Let's take the set of even numbers.

 

Is 2 = -2 ? No. So, Option A

Is 3 = 6? No. So, Option B //Note that here, $3^2=3+3=6$

Is set of even numbers finite? No. So, Option D

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