Question

1. Prove that in a group of even number of elements, at least one element, apart from
the identity element, must equal to its identity.​

Answer 1

Answer:

Define an equivalence relation on the set G underlying your group by declaring x∼y iff either x=y or x=y−1 . This partitions G into equivalence classes of size 1 and 2, so the order of G is a sum of 1's and 2's. Since the equivalence class of the identity e has size 1 and the order of G is even, there must be another equivalence class of size 1. The element in that class is its own inverse, so its square is the identity.

(Cauchy's theorem also says that if a prime p divides the order of a group G then G has an element of order p , which is exactly what you're asking in the case p=2 . But Cauchy's theorem (and its stronger cousin the Sylow theorems) is overkill here.)

Step-by-step explanation:

If you think I'm right you can mark me as brainliest.

Answer 2

SOLUTION

TO PROVE

In a group of even number of elements, at least one element, apart from the identity element, must equal to its identity

PROOF

Let G be the group

Let a ∈ G

$\sf{Then \: \: o(a) = o( {a}^{ - 1} )}$

$\sf{If \: \: o(a) < 3 \: \: then \: \: a = {a}^{ - 1} }$

$\sf{If \: \: o(a) \geqslant 3 \: \: \: then \: \: a \: \: and \: \: {a}^{ - 1} \: \: distinct }$

and they form a pair of elements of G such that Each is the inverse of the other.

Let us consider all pairs of elements of the form $\sf{\: \: \{ \: x, {x}^{ - 1} \} \: } \: where \: \: o(x) \geqslant 3$

All such pairs can not exhaust the whole G, because G contains atleast one element ( the identity element) of order less than 3

Since the order of G is even, an even number of elements having order less than 3 lie outside the union of all pairs of the form

$\sf{\: \: \{ \: x, {x}^{ - 1} \} \: } \: where \: \: o(x) \geqslant 3$

Since G contains only one element of order 1, the number of elements of order 2 must be odd

Hence in particular G must contain at least one element of order 2

Hence the proof follows

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Prove that the inverse of the product of two elements of a group is the product of the inverses taken in the reverse order

https://brainly.in/question/22739109

2. Show that the set Q+ of all positive rational numbers forms an abelian group under the operation * defined by a*b= 1/2(...

https://brainly.in/question/21467351