dropdownClick =(value ) => {
            this.setState({searchGroup: value.name, elementSelected: value})
        }

( I tried inserting the setState block inside the onClick and it do not worck too )

    <InputGroup className="mb-2">
        <Input value={this.state.searchGroup} onChange={(e) => this.filtergroups(e.target.value)} />
        <InputGroupButtonDropdown addonType="append" isOpen={this.state.drop} toggle={this.AbrirCerrarDropdown} >
        <DropdownToggle caret className="red button">Grupos</DropdownToggle>
        <DropdownMenu>{this.state.skillGroupsFiltered.map((element, index) => (
            <DropdownItem key={index} onClick={() => this.dropdownClick(element)}>{element.name}</DropdownItem>
        ))}
        </DropdownMenu>
    </InputGroup>