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Consider the set $\Sigma^*$ of all strings over the alphabet $\Sigma = \{0, 1\}$. $\Sigma^*$ with the concatenation operator for strings

1. does not form a group
2. forms a non-commutative group
3. does not have a right identity element
4. forms a group if the empty string is removed from $\Sigma^*$
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Identity element for concatenation is empty string $\epsilon$. Now, we cannot concatenate any string with a given string to get empty string $\implies$ there is no inverse for string concatenation. Only other 3 group properties -- closure, associative and existence of identity -- are satisfied and hence, ans should be (a).
by Active (2.4k points)
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U are right
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why not inverse?
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Identity element for concatenation is empty string $\epsilon$. Now, we cannot concatenate any string with a given string to get empty string $\implies$ there is no inverse for string concatenation.
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ok. similar to division by zero. not defined. illogical. doesnt make sense.

thanks.
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think about concatinating  string with some any other string to get null string...you will not find any such string so there is no inverse thats it
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difference between right identity and left identity element ?

There exists identity element e such that,

1) a*e = a for all a belongs to set    (right identity)

2) e*a = a for all a belongs to set     (left identity)

Is this correct ???
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Can we say Commutative property also fails here?
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of course Anil

$ab \neq ba$

$ab$ is string that start with a and end with b , and have length 2 while $ba$ is a string that starts with b and ends with a and have length 2 , clearly both are different strings.
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Identity element for concatenation is empty string ϵ and that is not in group so can we say that there is no identity element in set so it is not even a monoid?
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nice logic @ mgrwt
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@Mk Utkarsh-No your identity element is present in the set $\sum^*$ as

$\sum^*=(0+1)^*$

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